KR102590974B1 - Detection of stochastic process windows - Google Patents

Abstract

Methods, systems, and computer-readable media for constructing a lithography tool for manufacturing a semiconductor device are disclosed. The method includes selecting a first variable, selecting a second variable, selecting at least one response variable that is a function of the first variable and the second variable, and determining measurement uncertainty for each response variable. , based on the measurement of the response variable and the measurement uncertainty for the response variable, determining a plurality of probabilities representing a plurality of indications that a plurality of points associated with the lithographic process meet the specification requirements for each response variable - a plurality of probabilities. The probabilities represent a process window, and configuring a lithography tool for manufacturing a semiconductor device based on the process window.

Description

Detection of stochastic process windows

Cross-reference to related applications

This application claims priority and the benefit of U.S. Application No. 17/472,335, entitled “Detection of Probabilistic Process Windows,” filed September 10, 2021, the entire disclosure of which is hereby incorporated herein by reference as if fully reproduced hereinafter. Incorporated by reference.

This application is also a continuation-in-part of U.S. Application No. 17/316,154, entitled “System and Method for Generating and Analyzing Roughness Measurements,” filed May 10, 2021, and filed on Dec. 12, 2019. It is a continuation of U.S. application Ser. No. 16/730,393, filed Dec. 30 and titled “System and Method for Generating and Analyzing Roughness Measurements,” filed Dec. 12, 2018 and titled “System and Method for Generating and Analyzing It is a continuing application of U.S. Application No. 16/218,346, entitled "Roughness Measurements" (now U.S. Patent No. 10,522,322), and is a continuation-in-part of U.S. Application No. 15/892,080, entitled "Edge Detection System," filed February 8, 2018. Priority is claimed (current U.S. Patent No. 10,176,966). This application is related to U.S. Provisional Application No. 62/739,721, filed October 1, 2018 and titled “System and Method for Generating and Analyzing Roughness Measurements” and filed May 31, 2018 and titled “System and Method for Removing Noise From Priority is further claimed for U.S. Provisional Application No. 62/678,866, entitled “Roughness Measurements.” Additionally, as a continuation of U.S. Patent Application Serial No. 16/218,346, this patent application claims priority to U.S. Provisional Application Serial No. 62/602,152, entitled “Edge Detection System,” filed April 13, 2017. All of these applications are hereby incorporated by reference as if fully reproduced below.

This disclosure relates generally to edge detection of patterned structures, and more specifically, to images formed when using a scanning electron microscope (SEM) or other imaging device that produces images that contain undesirable noise. relates to edge detection of pattern structures in noise-prone images, and more specifically to analyzing these roughness measurements as a function of process variations and It is about optimizing processes and controlling process tools.

Generally, this disclosure provides methods, systems, and computer-readable media for generating a probabilistic process window that accounts for measurement uncertainty.

One aspect of the present disclosure includes a computer-implemented method. The method may include selecting a first process variable displayed on a first axis of the graph. The method may also include selecting a second process variable displayed on the second axis of the graph. The method may also include selecting at least one response variable that is a function of the first variable and the second variable. The method can also determine the measurement uncertainty for each response variable. The method may also provide a plurality of indications of whether a plurality of points associated with a lithographic process meet specification requirements for each response variable, based on a measurement of the response variable and a measurement uncertainty for the response variable. It may include determining probabilities, and the plurality of probabilities represent a process window. The method further includes configuring a lithography tool for manufacturing a semiconductor device based on the process window.

Another aspect of the present disclosure includes, in one implementation, a system that includes a lithography tool, a memory device storing instructions, and a processing device. The processing device is coupled to a memory device and a lithography tool. The processing device can execute instructions to select a first variable that can be displayed on a first axis of the graph. The processing device may also execute instructions to select a second variable that may be displayed on a second axis of the graph. The processing device may also execute instructions to select a response variable as a function of the first variable and the second variable. The processing device may also execute instructions to determine measurement uncertainty for a response variable. The processing device also includes instructions for determining a plurality of probabilities indicating a plurality of indications of whether a plurality of points associated with the lithography process meet specification requirements, based on the measurement of the response variable and the measurement uncertainty for the response variable. It can be run. A plurality of probabilities may represent a process window. The processing device may also execute instructions to configure a lithography tool for manufacturing a semiconductor device, based on the process window.

A further aspect of the disclosure includes a tangible, non-transitory computer-readable medium storing instructions that, when executed, cause a processing device to select a first variable displayed on a first axis of a graph. . The instructions may also cause the processing device to select a second variable displayed on the second axis of the graph. The instructions may also cause the processing device to select a response variable as a function of the first variable and the second variable. The instructions may also cause the processing device to determine measurement uncertainty for the output response variable. The instructions also cause the processing device to determine, based on the measurement of the response variable and the measurement uncertainty for the response variable, a plurality of probabilities indicating a plurality of indications of whether a plurality of points associated with the lithography process meet specification requirements. can do. A plurality of probabilities represent a process window. The instructions also cause the processing device to configure a lithography tool to fabricate a semiconductor device, based on the process window.

The attached drawings illustrate only exemplary implementations of the present disclosure and thus do not limit its scope since the inventive concept is suitable for other equally effective implementations.
1A is a representation of a pattern structure showing parallel line features with spaces between the lines.
1B is a representation of a pattern structure including contact hole features.
Figure 2 shows four different rough edges, all with the same standard deviation.
Figure 3 is a representation of power spectral density (PSD) versus frequency on a log-log scale.
Figure 4 is a graphical representation showing the roughness parameters PSD(0), correlation length, and roughness index plotted as power spectral density (PSD) versus frequency.
Figure 5 shows two power spectral densities (PSDs) corresponding to individual edges of a feature on the pattern structure.
Figure 6 is a graphical representation of the tradeoff of feature-to-feature variation and within-feature variation as a function of line length.
FIG. 7 is a block diagram of a scanning electron microscope (SEM) coupled to an information handling system (IHS), which together form one implementation of the disclosed edge detection device.
Figure 8A is a representation of a feature disposed on a substrate showing an electron beam impinging on the center of the feature.
FIG. 8B is a representation of a feature disposed on a substrate showing an electron beam impinging on the feature near its edge.
Figure 9 shows a grayscale image representation on top with the corresponding grayscale line scan along one horizontal cut graphically plotted immediately below.
Figure 10 shows an example of a patterned structure including a feature located on top of a substrate where the number of electrons escaping from the patterned structure varies depending on where the electron beam impinges on the patterned structure.
Figure 11 shows a predicted line scan of a resist step on a patterned structure, such as a silicon wafer.
Figure 12 shows another representative predicted line scan of a pattern of resist lines and spaces on a silicon wafer.
Figure 13a is an original SEM image of the pattern structure without using the disclosed edge detection device and method.
FIG. 13B is the same SEM image as FIG. 13A except that the disclosed edge detection device and method are used.
14 shows raw (biased) results both from the prior art (using a filter with conventional critical edge detection) and from using an inverse linescan model (ILM) without a filter. (Biased)) Linewidth roughness plot versus threshold settings.
Figure 15A is a plot of power spectral density (PSD) versus frequency of the right and left edges of a feature depicted before noise subtraction.
Figure 15B is a plot of power spectral density (PSD) versus frequency of the right and left edges of the feature depicted after noise subtraction.
Figure 16 shows portions of three SEM images of nominally identical lithography features taken at different SEM electron doses.
Figure 17A shows a typical line scan of a line feature on a wafer for the case where there are extremely large numbers of electrons so that pixel noise is negligible.
Figure 17b shows the 1-sigma uncertainty in edge detection location for perfectly smooth features in the presence of grayscale noise, for three different X pixel sizes.
FIG. 17C shows grayscale images as an example of using a simple critical edge detection algorithm that uses image filtering in the right image and no image filtering in the left image.
Figure 18 is a plot of linewidth roughness (LWR) PSD versus frequency showing the impact of two different image filters on a set of 30 images.
19 is a power spectral density plot versus frequency illustrating the noise subtraction process of the disclosed edge detection apparatus and method.
Figure 20 shows the PSD of a specific resist feature type on a given wafer, measured with different integrated frames in SEM.
Figure 21 shows biased and unbiased values of 3σ linewidth roughness (LWR) measured as a function of the number of integrated frames in the SEM.
Figure 22A shows biased linewidth roughness (LWR) power spectral densities (PSD) as a function of different pixel sizes and magnifications employed by the SEM.
Figure 22b shows unbiased linewidth roughness (LWR) power spectral densities (PSD) as a function of different pixel sizes and magnifications employed by the SEM.
FIG. 23 is a flow chart illustrating a representative overall process flow employed by the disclosed SEM edge detection system to detect edges of a pattern structure.
FIG. 24A is a grayscale representation of the pattern structure of vertical lines and spaces that the disclosed metrology tool analyzes.
Figure 24b shows a single line scan at one Y-pixel location.
Figure 24C shows an averaged line scan produced by averaging across all Y-pixels.
Figure 25A shows a PSD containing high frequency spike artifacts.
Figure 25B shows the PSD with spike artifacts removed.
Figure 26 shows a PSD containing mid-frequency spike artifacts and harmonics.
Figure 27A illustrates the impact of mid-frequency spike artifacts on modeling and interpretation of PSD.
Figure 27b shows the impact of removing mid-frequency spike artifacts on modeling and interpretation of PSD.
Figure 28A shows a PSD dataset showing a type of bump behavior.
Figure 28b shows an additional PSD dataset showing types of bump behavior.
Figure 29A shows modeling and analysis of a type I low frequency bump.
Figure 29b shows modeling and analysis of type II low frequency bumps.
Figure 30 is a flow diagram illustrating a representative process flow for detecting unwanted spikes in a PSD dataset, removing spikes from the PSD dataset and obtaining roughness parameters for a feature.
31 is a flow diagram illustrating another representative process flow for modeling bumps in a PSD dataset and obtaining unbiased roughness parameters for the feature.
Figure 32 is a plot of an example of a Gaussian distribution of mean values.
Figure 33 is a diagram showing an example of a heat-map indicating the probability that a plurality of points exist within a specification.
Figure 34 is an alternative three-dimensional diagram of the heat map of Figure 33.
Figure 35 is a diagram of an example heatmap illustrating fractions of specification features.
Figure 36 is an alternative three-dimensional diagram of the heatmap of Figure 35.
Figure 37 is a plot of an example curve for focus error and exposure error.
Figure 38 is a flow chart depicting a representative process flow for creating a probabilistic process window that accounts for measurement uncertainty.
Figure 39A is a diagram of an example of a Bossung plot combining the effects of focus and exposure dose on CD.
Figure 39B is an example of a contour plot containing a dimensional data set with contours of constant linewidth versus focus and exposure.
Figure 39C plots CD (nominal +/- 10%), 80 degree sidewall angle, and 10% resist loss as a function of focus (x-axis) and exposure dose (y-axis), all on the same graph. This is an example.
Figure 40A is an example diagram of a process window showing the two largest rectangles fitted inside the process window.
Figure 40B is an example diagram of a process window showing one maximum rectangle and one maximum ellipse fit inside the process window.
Figure 41 is an example analysis of a process window to provide exposure latitude versus depth of focus.
Figure 42 is an example of overlapping process windows for line/space patterns of two different pitches.
Figure 43 illustrates the potential impact of measurement error on geometric analysis of process window size.

Measuring the roughness of a pattern is complicated in that the noise in the measurement system is difficult to distinguish from the roughness being measured. It is common to use an imaging tool, such as a microscope, to create a detailed image of the object to be measured and then analyze information about that image to measure and characterize the roughness of one or more features of the object. In this case, noise in the acquired image may appear as roughness of features in the image. In order to generate more accurate measures of the roughness of features, techniques useful for separating the noise in the image from the actual roughness of the features, among others, are described below.

For example, scanning electron microscopes (SEMs) are very useful for studying features of patterned structures, such as semiconductor devices. Unfortunately, measuring the feature roughness of these structures is often challenging due to the noise inherent in SEM images. Filtering (smoothing) of the SEM image is typically necessary to achieve accurate edge detection, but such filtering undesirably changes the measured feature roughness. There is a need for an edge detection approach that reliably detects edges in very noisy SEM images without the use of image filtering (or at least without any filtering that would change the feature roughness being measured).

Pattern roughness is a major problem in many fields. Many, but not all, techniques for creating patterns of various shapes create roughness on the edges of these patterns, at least at the near molecular scale, if not at the larger scale. For example, in advanced lithography for semiconductor manufacturing, especially extreme ultraviolet (EUV) lithography, but also in other lithographic methods, the roughness of printed and etched patterns can cause many negative effects. Reduction of roughness requires a better understanding of the sources of stochastic variation, which in turn requires better measurement and characterization of rough features. Prior art roughness measurement approaches suffer from severe bias because noise in the image adds to the roughness on the wafer. This disclosure provides a practical approach to making unbiased roughness measurements through the use of a physically based inverse line scan model. This enables accurate and robust measurement of roughness parameters over a wide range of SEM measurement conditions.

Before discussing implementations of the disclosed technology that address the SEM image noise problem, the present disclosure first discusses the frequency dependence of the lithography and roughness of patterned structures.

1. Stochastic effects in lithography

Lithography and patterning advances continue to drive Moore's Law by cost-effectively reducing the area of silicon consumed by transistors in integrated circuits. In addition to the need for improved resolution, these lithographic advances should also allow for improved control of the smaller features being manufactured. Historically, lithographs have improved patterning fidelity (e.g., exposure dose and focus variations, hotplate) by attempting to minimize sources of these variations and developing processes with minimal sensitivity to these variations. The focus was on “global” sources of variation that affect temperature inhomogeneities, scanner aberrations). However, today's small features also suffer "local" fluctuations caused by the underlying stochasticity of patterning near the molecular scale.

In lithography, light is used to expose a photosensitive material called photoresist. The resulting chemical reactions (including those that occur during post-exposure bake) change the solubility of the resist, allowing patterns to be developed and creating the desired critical dimension (CD). For a “large” volume of resist (i.e., a volume containing a large number of resist molecules), the amount of light energy averaged over that volume will produce a certain (average) amount of dissolution to produce the pattern. Produces (on average) a certain amount of chemical change. The relationships between light energy, chemical concentration, and dissolution rate can be described by deterministic equations that predict outputs for a given set of inputs. These lithography models are extremely useful and are commonly used to understand and control lithography processes for semiconductor manufacturing.

This deterministic view of the lithography process (certain inputs always produce certain outputs) is only approximately true. The "mean field theory" of lithography states that, on average, deterministic models accurately predict lithography results. If we average over a large number of photons, a single number for the light energy (average) is sufficient to describe the light energy. For large volumes of resist, the average concentration of a chemical species sufficiently describes its chemical state. However, for very small volumes, the number of atoms or molecules in the volume is random, even for a fixed "average" concentration. This randomness within small volumes (i.e., for small numbers of photons or molecules or events) is commonly referred to as “shot noise” and occurs when the region of interest approaches the molecular scale in lithography. This is an example of stochastic variation in .

A stochastic process is one in which the outcome of the process is determined randomly. At the atomic/molecular level, all processes are inherently stochastic. For semiconductor patterning at the 20-nm node and below (with minimum feature sizes of less than 40 nm), the dimensions of interest become significant and even affect the dimensions, shapes, and placement of the patterns that are fabricated. It is small enough to dominate the total fluctuations that affect it. These stochastic effects may also be important for larger feature sizes in some situations.

The most striking manifestation of stochastic fluctuations in lithography (as well as other parts of the etching and patterning process) is that the resulting patterns are rough rather than smooth (Figure 1a). In the pattern structure shown in Figure 1A, nominally parallel vertical lines appear as bright vertical areas, while spaces appear as dark vertical areas between the lines. The roughness of the feature edge is called line-edge roughness (LER), and the roughness of the feature width is called line width roughness (LWR). The roughness of the feature centerline (midpoint between the left and right edges) is called pattern placement roughness (PPR). Another important consequence of these stochastic variations is the random variation in the size, shape and placement of features, which is particularly evident for contact hole features (Figure 1b).

Stochastic effects in patterning can reduce the yield and performance of semiconductor devices in several ways: a) within-feature roughness affects the electrical properties of the device, such as metal line resistance and transistor gate leakage; can have an impact; b) feature-to-feature size variation (also referred to as local CD uniformity, LCDU) caused by stochasticities adds to the total budget of CD variation, sometimes becoming the dominant source; c) feature-to-feature pattern placement variation (also referred to as local pattern placement error, LPPE) caused by stochasticities adds to the total budget of PPE, sometimes becoming a dominant source; d) Rare events causing larger than expected occurrences of catastrophic bridges or breaks are more likely if error distributions have fat tails; e) Decisions based on metrology results (including process monitoring and control, as well as calibration of optical proximity correction (OPC) models) ensure that these metrology results do not adequately account for stochastic variations. In this case, it may be defective. For these reasons, proper measurement and characterization of stochastic-induced roughness is important.

Many other types of devices are also sensitive to feature roughness. For example, roughness along the edges of an optical waveguide can cause loss of light due to scattering. Feature roughness in radio frequency microelectromechanical systems (MEMS) switches can affect performance and reliability, as is true for other MEMS devices. Feature roughness can reduce the output of light emitting diodes. Edge roughness can also affect the mechanical and wetting properties of features in microfluidic devices. The roughness of features within a wire grid polarizer can affect the efficiency and transmission of the polarizer.

Unfortunately, prior art roughness measurements (e.g., measurement of linewidth roughness or line edge roughness using a critical dimension scanning electron microscope (CD-SEM)) are susceptible to measurement noise caused by the measurement tool. polluted by This results in biased measurements, where measurement noise is added to the true roughness, creating an apparent roughness that overestimates the true roughness. Moreover, this bias depends on the specific measurement tool used and its settings. These biases are also a function of the pattern being measured. Prior art attempts to provide unbiased roughness estimates often suffer in many today's applications due to smaller feature sizes and higher levels of SEM noise.

Accordingly, a new approach is needed to make unbiased roughness measurements that avoid the problems of prior art attempts and provide an accurate and precise unbiased estimate of feature roughness. Additionally, a good pattern roughness measurement method should have minimal dependence on metrology tool settings. CD-SEM settings such as magnification, pixel size, number of averaging frames (equivalent to the total electron dose in the SEM), voltage and current can cause quite large changes in the measured biased roughness. Ideally, unbiased roughness measurements would be independent of these settings to a large extent.

2. Frequency dependence of line edge roughness (LER), linewidth roughness (LWR) and pattern placement roughness (PPR).

Coarse features are most commonly characterized by edge position (for LER), linewidth (for LWR), or standard deviation of the feature centerline (for PPR). However, accounting for the standard deviation is not sufficient to fully account for roughness. Figure 2 shows four different rough edges, all with the same standard deviation. The significant differences seen in the edges make it clear that the standard deviation is not sufficient to fully characterize the roughness. Instead, a frequency analysis of the roughness is required. The four randomly rough edges shown in Figure 2 all have the same standard deviation of roughness, but differ in the frequency parameters of the correlation length (ξ) and roughness index (H). More specifically, referring to Figure 2, in case a) ξ=0, H=0.5; b) case ξ=10, H=1.0; c) case ξ=100, H = 0.5; and d) case ξ= 0.1, H = 0.5.

The standard deviation of the rough edge is perpendicular to the ideal straight line and represents the variation therewith. In Figure 2, the standard deviation represents the vertical variation of the edge. However, the fluctuations may spread differently along the length of the line (in the horizontal direction in Figure 2). This line-length dependence can be described using a correlation function, such as an autocorrelation function or a height-to-height correlation function.

Alternatively, the frequency f can be defined as one over the length along the line (Figure 3). The dependence of roughness on frequency can be characterized using the well-known power spectral density (PSD). PSD is the variance of the edge per unit frequency (Figure 3) and is calculated as the square of the coefficients of the Fourier transform of the edge variation. The low frequency region of the PSD curve describes edge fluctuations that occur over long length scales, while the high frequency region describes edge fluctuations over short length scales. Typically, PSDs are plotted on the log-log scale used in FIG. 3.

The PSD of a lithographically defined feature generally has a shape similar to that shown in Figure 3. The low-frequency region of the PSD is flat (so-called "white noise" behavior) and then falls off with the power of the frequency above a certain frequency (statistically, fractal behavior). The difference between these two areas is related to the correlation along the length of the feature. Points along distant edges are uncorrelated (statistically independent), and uncorrelated noise has a flat power spectral density. However, at short length scales, edge deviations are correlated, reflecting correlated mechanisms in the generation of roughness, such as acid reaction-diffusion for chemically amplified resists. The transition between uncorrelated and correlated behavior occurs at a distance called the correlation length.

Figure 4 shows that a typical PSD curve can be described with three parameters. PSD(0) is the zero-frequency value of PSD. Although this value of PSD can never be measured directly (zero frequency corresponds to an infinitely long line), PSD(0) can be considered as the value of PSD in the flat low frequency region. The PSD begins to fall off near a frequency of 1/(2πξ), where ξ is the correlation length. In the fractal domain, we have what is sometimes referred to as "1/f" noise, and the PSD has a slope (on a log-log plot) corresponding to the power of 1/f. The slope is defined as 2H + 1, where H is called the roughness exponent (or Hurst exponent). Typical H values are between 0.5 and 1.0. For example, H = 0.5 when a simple diffusion process causes the correlation. Each of the parameters of the PSD curve has important physical implications for the lithographically defined feature, as discussed in more detail below. The variance of roughness is the area under the PSD curve and can be derived from the other three PSD parameters. The exact relationship between the variance and the other three PSD parameters depends on the exact shape of the PSD curve in the mid-frequency region (defined by the correlation length), but an approximate relationship represents a general trend according to equation 1 below: Can be used:

(One)

The differences observed in the individual four rough edges of Figure 2 can now be easily understood as differences in the PSD behavior of the features. Figure 5 shows two PSDs corresponding to edges a) and edge c) from Figure 2. These two edges have the same variance (same area under the PSD curve), but they have different values of PSD(0) and correlation length (in this case the roughness index remains constant). Although the standard deviation of the roughness of edge a) and edge c) is the same, these edges show different PSD behavior. As discussed below, different PSD curves will result in different roughness behavior for finite length lithographic features.

3. Influence of frequency behavior of roughness

The roughness of the lines and spaces of the pattern structures is characterized by measuring very long lines and spaces, long enough for flat areas of the PSD to become apparent. For sufficiently long features, the measured LWR (i.e., the standard deviation σ of the linewidths measured along the line) can be thought of as the LWR of an infinitely long feature, σ _LWR (∞). However, patterned structures, such as semiconductor devices, are made of features with varying lengths L. For these shorter features, the stochasticities give rise to the standard deviation of the average linewidths of the features, the feature-to-feature variation explained by σ _CDU (L), and the within-feature roughness σ _LWR (L). something to do. These feature-to-feature variations represent local critical dimension (CD) variations that are not caused by well-known “global” error sources (scanner aberrations, mask illumination non-uniformity, hot plate temperature variations, etc.). It is referred to as local critical dimension uniformity (LCDU).

For a line of length L, the within-feature variation and feature-to-feature variation are bound to an infinitely long (of the same nominal CD and pitch) by the Conservation of Roughness principle given in Equation 2 below: It can be related to the LWR of the line:

(2)

The roughness preservation principle states that the variance of very long lines is partitioned into a within-feature variance and a feature-to-feature variance for shorter lines. How this partition occurs is determined by the correlation length, or more specifically by L/ξ. Using the basic model for the shape of a PSD as an example, it is recognized as:

(3)

Thus, Equations 1-3 show that the measurement of PSD for a long line, and its description by the parameters PSD(0), ξ and H, allow predicting the stochastic impact for a line of arbitrary length L . Note that LCDU does not depend on the roughness exponent and places less importance on H than PSD(0) and ξ. For this reason, it is useful to describe the frequency dependence of roughness using an alternative triplet of parameters: σ _LWR (∞), PSD(0), and ξ. Note that these same relationships apply to LER and PPR.

Also, looking at Equation 3, note that the correlation length is a length scale that determines whether a line of length L acts "long" or "short." For long lines, L >> ξ and the local CDU operates according to equation 4 below:

(4)

This long line result provides a useful interpretation of PSD(0): the square of the LCDU for a given line multiplied by the length of that line. Decreasing PSD(0) by a factor of 4 reduces LCDU by a factor of 2, and other PSD parameters have no effect (as long as L >>ξ). Typically, resists yielded correlation lengths on the order of one-quarter to one-half the minimum half-pitch of their lithographic production. Therefore, it is generally in this long line length regime when the features are longer than approximately 5 times the minimum half pitch of the technology node. For shorter line lengths, correlation length also starts to become important.

Equation 1-3 represents the trade-off of within-feature variation and feature-to-feature variation as a function of line length. Figure 6 shows an example of this relationship. For very long lines, the LCDU is small and the intra-feature roughness approaches its maximum value. For very short lines, LCDUs are superior. However, due to the quadratic nature of conservation of roughness, σ _LWR (L) rises very quickly as L increases, while LCDU falls very slowly as L increases. Therefore, there is a wide range of line lengths where both feature roughness and LCDU are significant.

Since the roughness preservation principle also applies to PPR, short features suffer from local pattern placement errors (LPPE) as well as local CDU problems. For the case of uncorrelated left and right edges of a feature, the PSD(0) for LWR is typically twice the PSD(0) for LER. Likewise, the PSD(0) of LER is typically twice the PSD(0) of PPR. Therefore, typically, LPPE is about half of LCDU. When the left and right feature edges are significantly correlated, this simple relationship no longer holds.

4. Roughness measurement of patterned structures using scanning electron microscopy (SEM)

A common method for measuring feature roughness for small features is top-down critical dimension scanning electron microscopy (CD-SEM). A typical light microscope has a magnification of up to 1000X and a resolution of up to hundreds of nanometers. Scanning electron microscopes use electrons to create very small spots (around 1 nm wide) that can be used to create high-resolution images, with magnifications exceeding 20,000X. CD-SEM is an SEM optimized for measuring the dimensions of a wide range of features found on semiconductor wafers. They can measure the average critical dimension of coarse features with high precision, but have also proven to be very useful in measuring LER, LWR, PPR, and their PSD. However, there are errors in SEM images that can greatly affect the measured roughness and roughness PSD while having little effect on the measurement of average CD. For this reason, the metrology approach required for measuring PSD can be quite different from the approach commonly used for measuring average CD.

FIG. 7 shows a block diagram of one implementation of the disclosed edge detection system 700 that determines feature roughness. Pattern structure 800 and electronic imaging optics 710, 715, 720, and 725 are located within a vacuum chamber 703 that is evacuated by vacuum pump 702. Electrons are generated from a source such as an electron gun 705 to form an electron beam 707. Typical electron beam sources include a heated tungsten filament, a lanthanum hexaboride (LaB6) crystal formed with a thermionic emission gun, or a sharp-tipped metal wire formed to create a field emission gun. The emitted electrons are accelerated and focused using electromagnetic condenser lenses 710, 715, and 720. The energy of the electrons striking the patterned structure 800 typically ranges from 200 eV to 40 keV for SEM, but more typically 300 eV to 800 eV for CD-SEM. The final condenser lens 720 uses a scanning coil 725 to provide an electric field that deflects the electron beam 707 toward the patterned structure 800 as a focus spot. The scan coil 725 scans a focused spot across the pattern structure 800 through the final lens aperture 735 in a raster scan fashion to expose a specific field of view on the pattern structure 800. The SEM 701 includes a backscattered electron detector 740 that detects backscattered electrons back from the patterned structure 800 . SEM 701 also includes a secondary electron detector 745 as shown in FIG. 7 . Before imaging patterned structure 800, a user places patterned structure 800 on patterned structure receptor 732, which supports and positions patterned structure 800 within SEM 701. SEM 701 includes a controller (not shown) that controls raster scanning of patterned structure 800 during imaging.

Referring now to FIGS. 8A and 8B, electrons of electron beam 707 impinging on patterned structure sample 800 undergo a number of processes depending on the electron energy and material properties of the sample. Electrons scatter atoms of the sample material, releasing energy, changing direction, and often ionizing the sample atoms, creating a cascade of secondary electrons. Some of these secondary electrons 805 may escape from the pattern structure 800, and others may remain inside the pattern structure. Pattern structure 800 includes a substrate 810, such as a semiconductor wafer. Features 815 are disposed on substrate 810 as shown in Figure 8A. Features 815 may be metal lines, semiconductor lines, photoresist lines, or other structures on substrate 810. Features 815 may have other shapes, such as pillars or holes, or more complex shapes. Feature 815 may be repeated or isolated from other features on the pattern structure. The space surrounding features 815 can be empty (vacuum or air) or filled with different materials. Patterned structure 800 may be a liquid crystal or other flat panel display, or other patterned semiconductor or non-semiconductor device. Feature 815 includes edges 815-1 and 815-2. The area of feature 815 where electron beam 707 interacts with feature 815 may exhibit a tear-droplet-like shape, for example, as shown in FIG. 8A. is the working volume (820).

Sometimes electrons bounce back from the atomic nucleus and escape the sample (called backscattered electrons). Some of the lower energy secondary electrons 805 may also escape out of the sample 800 (often through the edges of the feature, see FIG. 8B). The way an SEM forms an image is by detecting, for each beam position, the number of secondary and/or backscattered electrons escaping the sample.

As the electron beam is scanned across the patterned structure 800 during one line scan, it “dwells” in a specific spot at a specific time. During that dwell time, the number of electrons detected by the backscattered electron detector 740 or the secondary electron detector 745, or both, is recorded. The spot is then moved to the next "pixel" location, and the process is repeated. The result is a two-dimensional array of pixels (positions along the surface of the sample) with the detected electron counts digitally recorded for each pixel. The count is then typically normalized and expressed as an 8-bit grayscale value between 0 and 255. This allows the detected electron counts to be plotted as grayscale “images” such as these images shown in Figure 1. It is important to note that while the images coming from the SEM remind the viewer of the optical images perceived through the eye, these grayscale images are actually just convenient plots of the collected data.

CD-SEM uses SEM images to measure the width of features. The first step in measuring feature width is detecting the edges of the features. For pixels near the edge of a feature, a greater number of secondary electrons escape through the feature edge, creating bright pixels called “edge bloom” (see FIGS. 8B and 9). It is this bright edge bloom that allows detection of feature edges. For example, in the grayscale image representation in the top portion of Figure 9, such edge blooms are observed at edges 905 and 910 of feature 915. A line scan is essentially a horizontal cross section through a 2D SEM image that provides grayscale values as a function of horizontal pixel position on the feature, as in the graph shown in the bottom part of Figure 9.

Data from a single horizontal row of pixels across the sample is referred to as a “linescan.” Note that the term linescan is used broadly to include cases where an image is formed without the use of scanning. The positions of the edges of a feature can be detected from a single linescan, or from a collection of linescans representing the entire image shown in the top portion of Figure 9. These same edges appear as peaks 905' and 910' in the grayscale value vs. pixel position graph in the bottom portion of Figure 9. Once the edges of a particular feature are determined, the width of a particular feature is the difference between the positions of these two edges.

5. Line scan model

Images are created through a physical process based on a microscope or other imaging tool used to acquire an image of the structure. Often these images are two-dimensional arrays of data, where an image can be thought of as a set of data derived from a structure. A single, one-dimensional cross section through the image is called a line scan. The imaging tool's model can predict the image for a given structure being imaged. For example, a model that describes scanning electron microscopy can predict the images that will be acquired by an SEM when imaging a given structure.

CD-SEM converts a measured line scan, or series of measured line scans, into a single dimensional number, the measured CD. To better understand how line scans relate to the actual dimensions of the feature being measured, it is important to understand how the systematic response of the SEM measurement tool to pattern structures affects the shape of the resulting line scans. . Rigorous 3D Monte Carlo simulations of SEM line scans can be very valuable for this purpose, but they are often too computationally expensive for routine use. Therefore, one approach is to develop a simplified analytical linescan model (ALM) that is computationally more suitable for the task of rapidly predicting line scans. ALM adopts the physics of electron scattering and secondary electron generation, and each term in the model has a physical meaning. This analytical line scan representation can be amenable to rigorous Monte Carlo simulations to validate and calibrate its use.

A common application for ALM was a classic forward modeling problem: given the material properties (for the feature and the substrate) and the geometric description of the feature (width, pitch, sidewall angle, top corner rounding, footing, etc.), ALM Predict the resulting line scan. The mathematical details of ALM are found in the publication: Chris A. Mack and Benjamin D. Bunday, “Analytical Linescan Model for SEM Metrology”, Metrology, Inspection, and Process Control for Microlithography XXIX, Proc., SPIE Vol. 9424, 94240F (2015), and Chris A. Mack and Benjamin D. Bunday, “Improvements to the Analytical Linescan Model for SEM Metrology”, Metrology, Inspection, and Process Control for Microlithography XXX, Proc., SPIE Vol. 9778, 97780A (2016), the disclosures of both publications are hereby incorporated by reference in their entirety. Other models with similar inputs and outputs may also be used.

The analytical line scan model (ALM) is briefly reviewed below. Mathematical modeling shows that the interaction of an electron beam with a flat sample of a given material is a double Gaussian with a forward scattering width and a fraction of the energy deposited by these backscattered electrons, and a backscattering width and the fraction of energy deposited by these backscattered electrons. We begin by assuming that we are generating an energy stacking profile that takes the form of a Gaussian. This model also states that the number of secondary electrons generated within the material is directly proportional to the energy deposited per unit volume, and that the number of secondary electrons escaping the wafer (and thus detected by the SEM) is related to the secondary electrons near the top of the wafer. It is assumed that it is directly proportional to the number of electrons.

Secondary electrons that reach the detector will appear at a certain distance r from the location of the incident beam. From the above assumptions, the number of secondary electrons detected will be a function given in equation 5 below:

(5)

Here, σ _f and σ _b are the forward and backscattering ranges, respectively, and a and b are the amounts of forward and backscattering, respectively.

SEMs detect topography due to the different number of secondary electrons escaping when the beam is in the space between features compared to when it is above the features. Figure 10 shows that secondary electrons have trouble escaping from space (especially when it is small), making the space appear relatively dark. When the electron beam is focused on a spot in the space between the lines, the scattered electrons interact with feature 815 absorbing some of the escaping secondary electrons. The detected secondary electron signal decreases as the beam approaches a feature edge in space.

Absorption by steps (i.e., features 815) can be modeled to generate a prediction of the shape of the line scan in the spatial domain. If the large feature has a left edge 815-1 at x = 0, and feature 815 is on the right (positive x), then the detected secondary electron signal (SE(x)) as a function of position is It will be given by equation 6:

(6)

Here, α _f is the fraction of forward-scattered secondary electrons absorbed by the step, and α _b is the fraction of back-scattered secondary electrons absorbed by the step.

However, when the beam is on top of feature 815, the interaction of the scattered electrons with the feature is very different, as explained in Equation 7 below. As illustrated in Figure 8, two phenomena occur when the beam is closer to the edge than further away. First, secondary electrons from both forward and back scattered electrons can more easily escape from edge 815-1. This causes the edge bloom already discussed above. To account for this effect, a positive term is added, accounting for the enhanced escape of forward-scattered secondary electrons, where σ _e is very similar to the forward-scattering range of the step material. Additionally, the interaction volume itself is reduced when the beam is near edge 815-1, resulting in fewer secondary electrons. Therefore, if σ _v < σ _e The terms are subtracted to give equation 7 below, which is a line scan representation of the top of the large feature 815:

(7)

Figure 11 shows an example of the results for this model. More specifically, Figure 11 shows a predicted line scan of a left-facing resist step 815 (a large feature with left edge 815-1 at x = 0) on a substrate, such as a silicon wafer. The calibrated model 1105 is overlaid on the rigorous Monte Carlo simulation results 1110. The calibrated model 1105 matches the Monte Carlo simulation results 1110 so closely that the two curves appear together almost as one line.

The above discussion includes modeling an isolated left edge 815-1. Adapting the model to include a right-facing edge involves translating and inverting the edge and adding the resulting secondary (i.e., secondary electrons). If two edges are close enough to interact, some complications arise and additional terms are created. Additionally, the influence of non-vertical sidewalls and rounded corners at the top and bottom of the feature edge can be included in the model (FIG. 12).

Figure 12 shows a representative predicted linescan of a pattern of resist lines and spaces on a silicon wafer. The calibrated model 1205 is overlaid on the rigorous Monte Carlo simulation results 1210. Again, the calibrated model 1205 matches the Monte Carlo simulation results 1110 so closely that the two curves appear together almost as one line. The final model (ALM) includes 15 parameters that depend on the properties of the materials of the feature and wafer, and the beam voltage. To verify the model and calibrate these parameters, rigorous first principles Monte Carlo simulations can be used to generate line scans for different materials and feature geometries. The ALM can then be fitted with Monte Carlo results, producing best-fitting values for the 15 unknown parameters.

6. Inverse Linescan Model

Line scan or image models, such as the analytical line scan model (ALM) discussed above, predict the shape of an image or image line scan for a particular pattern structure (eg, a feature on a wafer). ALM solves the forward modeling problem in which the model receives geometric structure information for a specific feature as input and provides the predicted shape of an individual SEM line scan of that specific feature as output.

In contrast to ALM, the disclosed edge detection system 700 includes a reverse model that receives as input “measured linescan information” from the SEM 701 that describes specific features on the wafer. In response to measured line scan information describing a particular feature, edge detection system 700 is configured to generate output “feature geometry information” that describes the feature geometry that will produce the measured line scan. We adopt his reverse model. Advantageously, edge detection system 700 has been found to be effective even when line scan information measured from SEM 701 contains a significant amount of image noise. In one implementation, the output feature geometric information includes at least the feature width. In other implementations, the output feature information may include feature width and/or other geometries relative to the geometry of a particular feature, such as sidewall angle, feature thickness, top corner rounding, or bottom footing. Includes descriptor . Note that the features placed on the semiconductor wafer are examples of one specific type of pattern structure to which the disclosed technology applies.

Like many models of imaging systems, ALM is inherently nonlinear. To address the nonlinear nature of ALM, edge detection system 700 numerically inverts the ALM or similar forward model and fits the resulting inverse line scan model to the measured line scan to detect feature edges (e.g. (e.g., to estimate feature geometry on the wafer). The disclosed edge detection system apparatus and edge detection process include the ability to detect and measure feature roughness. The disclosed apparatus and methodology can also be applied to other applications of general CD metrology of 1D or 2D features, such as precise measurement of feature width (CD) and edge position or placement.

First, note that ALM (and similar models as well) has two types of input parameters: material-dependent parameters and geometry parameters. Material-dependent parameters include parameters such as forward and backscattering distances, while geometric parameters include parameters such as feature width and pitch. In one implementation, for repeated edge detection applications, the material parameters will be fixed and only the geometry parameters will vary. In the simplest case (i.e., for simple edge detection), only the edge positions for the feature are assumed to be changing, so the sidewall angle, corner rounding, etc. are assumed to be constant. Accordingly, use of the line scan model for edge detection in edge detection system 700 involves two steps: 1) calibrating parameters that are assumed to be constant across the entire image, and then 2) performing a line scan for each measurement. This involves finding feature edge positions that provide the best fit of the measured line scan to the model.

In one implementation, in the first step, calibration is achieved by comparing the line scan model to rigorous Monte Carlo simulations. The goal of this step is to find material parameters across the required range of applications and ensure that the fitting is appropriate for the required range of feature geometries. When complete, this calibrated line scan model can serve as a starting point for the creation of an inverse line scan model. The Inverse Linescan Model (ILM) must be calibrated to the specific SEM image to be measured. Because image grayscale values are only proportional to the secondary electronic signals, minimal mapping to grayscale values is required. In real-world applications, the material properties in experimental measurements will not be the same as those assumed in Monte Carlo simulations, so some calibration of these parameters will also be required.

7. Calibration of the inverse line scan model

Before using the ILM for edge detection, the ILM is first calibrated. Some parameters of the model (e.g., material-dependent parameters) are assumed to be constant for the entire image. However, geometric parameters such as edge positions, feature widths, and pitches are assumed to vary from line scan to line scan. The goal of ILM calibration is to determine parameters that are constant for the entire image, regardless of the exact location of feature edges. Accurate determination of these parameters in the presence of image noise is a further goal of ILM calibration. These goals are achieved by averaging along the axis of symmetry for the feature being measured, thus averaging both image noise and actual feature roughness.

By averaging the line scan along an axis of symmetry (eg, a direction parallel to a long line or spatial feature), information about the actual edge positions is lost but information about the material parameters of the line scan model is maintained. Additionally, most of the noise in the image is averaged out in this way. Calibrating the ILM to the average line scan can create a set of material parameters specific to this image (or arbitrary parameters assumed to be constant throughout the image).

Many of the features to be measured represent axes of symmetry that are suitable for ILM calibration. For example, a vertical edge has a vertical axis of symmetry. Averaging all pixels in a vertical column of pixels from an image will average out all vertical variation, leaving only the horizontal information, in the direction perpendicular to the edge of the feature. The result of this averaging is a one-dimensional line scan, called an average line scan. Likewise, nominally circular contact holes or pillars are ideally radially symmetric. Averaging over the polar angle about the center of the feature will produce an average line scan that removes noise and roughness from the image. Oval hole shapes can be so averaged by compressing or expanding the pixel size in one direction proportional to the ratio of the major axis to the minor axis of the ellipse. Other axes of symmetry exist for other features as well.

One measured image (eg, one SEM image) may include one or more features within the image. For example, Figure 1A shows multiple vertical line features and multiple vertical space features. Figure 1B shows multiple contact holes. For such cases, each feature can be individually averaged along the axis of symmetry to form an average line scan for that feature. In the example of Figure 1A, the SEM image can be partitioned into vertical stripes, each stripe containing only one line feature, where the stripe extends from approximately the center of one space to approximately the center of the next space. extends horizontally. For the example of Figure 1B, the image may be partitioned into separate rectangular regions, each containing exactly one contact hole with the center of the contact hole approximately coinciding with the center of the rectangular region. The averaged line scan for that contact hole is then determined from that rectangular region of the image. Alternatively, each averaged line scan from each feature within the image can itself be averaged together to form a single averaged line scan applicable to the entire image.

For repeated edge detection applications (such as detection of all edges on a single SEM image), the material parameters will be fixed and only the geometry parameters will vary. In the simplest case (i.e. for simple edge detection), only the edge positions for the feature can be assumed to be changing, so the feature thickness, sidewall angle, corner rounding, etc. are assumed to be constant. Therefore, the use of ILM for edge detection will involve two steps: calibrating once for parameters assumed to be constant (i.e. material and fixed geometric properties) using an average line scan; and then finding feature edge positions that provide the best fit of the measured line scan to the line scan model for each line scan. Optionally, calibration is achieved by comparison of a line scan model with a rigorous Monte Carlo simulation, as previously described. The goal of this initial step is to find material parameters across the required range of applications and ensure that the model is appropriate for the required range of feature geometries. When complete, this partially calibrated line scan model must continue to be fully calibrated for the specific SEM image to be measured using average line scan.

Once the ILM has been calibrated with a given SEM image or sets of images, it is used to detect edges. Due to the non-linear nature of line scan models such as ALM models, for example, using non-linear least-square regression to find the values of the left and right edge positions that best fit the model to the data. Numerical inversion is required. For simpler line scan models, a linear least-square fit may be possible. Other means of “best fit” are also known in the art. ILM as an edge detector allows edge detection in a high-noise environment without using a filter. Figures 13a and 13b demonstrate reliable detection of edges for a very noisy image without using any filtering or image smoothing. More specifically, Figure 13a is the original SEM image of the patterned structure showing 18 nm lines and spaces before edge detection using ILM. Figure 13b is the same image after edge detection using ILM.

The Gaussian filter is a common image smoothing filter designed to reduce noise in images. Other filters such as box filters and median filters are also commonly used for this purpose. To illustrate the impact of image filtering on roughness measurements, Table 1 below shows the measured 3σ linewidth roughness (LWR) as a function of Gaussian filter x- and y-width (in pixels). In each case, the ILM edge detection method was used, so the resulting difference in LWR is only a function of the image filter parameters. This range is almost a factor of 2, showing that many different roughness measurements can be obtained based on arbitrary selection of filter parameters. ILM edge detection was used in all cases. Using conventional threshold edge detection methods, the resulting range of 3σ roughness values is much larger (Table 2). Similar results were obtained when using other filter types (e.g. box or median).

Table 1

Raw (biased) 3σ LWR (nm) as a function of Gaussian filter x- and y-width (in pixels), using ILM edge detection.

Table 2

Raw (biased) 3σ LWR (nm) as a function of Gaussian filter x- and y-width (in pixels), using conventional critical edge detection.

The arbitrary choice of image filter parameters has a significant impact on the measurement of the roughness of the pattern structure, but the impact of the threshold depends on the specific edge detection method used. For the case of simple threshold edge detection after image filtering, there is one threshold that minimizes the measured roughness of 3σ; other values change the roughness quite dramatically (see Figure 14). For ILM, the choice of threshold has little effect on the measured LWR (in Figure 14, when the threshold is varied from 0.25 to 0.75, LWR varies from 5.00 nm to 4.95 nm). In prior art edge detection methods, arbitrary choice of threshold can cause large variations in measured roughness. For ILM, there are essentially no arbitrary choices that affect the measurement of roughness.

Although the disclosed ILM system achieves accurate detection of edges in the presence of high levels of noise, the noise still adds to the measured roughness. For a line scan of a given edge slope, the uncertainty of the grayscale values near the line edge translates directly to the uncertainty at the edge location. However, the main difference is that the impact of noise can be measured in that case without filtering. The noise floor of the unfiltered image can be subtracted from the PSD (power spectral density) to produce an unbiased estimate of the PSD (and therefore roughness). In the case of filtered images, the noise floor is largely smeared and cannot be detected, measured, or removed.

Figures 15A and 15B show LER power spectral densities from a number of coarse features with right and left edges individually combined. More specifically, Figure 15A shows raw PSDs after edge detection using the disclosed ILM technique, while Figure 15B shows PSDs after noise subtraction.

Consider the results shown in Figure 15A, where the line edge roughness (LER) for the left and right edges of a feature on the pattern structure are compared. The raw PSDs show that the two edges behave differently. However, these differences are an artifact of the SEM caused by scan-direction asymmetry (such as charging) that causes the right line scan slope to be lower than the left line scan slope. In fact, for this sample there is no difference between the right and left edges on the wafer. By measuring the noise floor for each edge separately, subtracting the noise produces a common left/right LER (Figure 15b), which is an unbiased estimate of the actual PSD.

Once the noise is subtracted, reliable analysis of the PSD can lead to reliable estimates of important roughness parameters such as zero frequency PSD(0), correlation length, and roughness index H. Unbiased 3σ roughness can also be obtained. Without removing noise, extraction of these parameters from the empirical PSD is problematic and prone to systematic errors.

8. Unbiased Measurement of PSD

The biggest obstacle to accurate roughness measurement is noise in CD-SEM images. Among other noise sources, SEM images suffer from shot noise, where the number of electrons detected for a given pixel varies randomly. For an expected Poisson distribution, the variance of the number of electrons detected for a given pixel in the image is equal to the expected number of electrons detected for that pixel. Because the number of detected electrons is proportional to the number of electrons impinging on the sample location represented by that pixel, the relative amount of noise can be reduced by increasing the electron dose received by the sample. For some types of samples, the electron dose may increase with several results. However, for other types of samples (eg, photoresist), high electron doses result in sample damage (eg, resist line slimming). Other types of samples, such as biological samples, can also experience electronic damage. Therefore, to prevent sample damage, the electron dose is kept as low as possible, where the lowest possible dose is limited by the noise in the resulting image.

Figure 16 shows portions of three SEM images of nominally identical lithographic features taken at different electron doses. More specifically, Figure 16 shows portions of SEM images of nominally identical resist features with integrations of 2, 8, and 32 frames (from left to right, respectively). Doubling the integration of frames doubles the electron dose per pixel. In each case, because the dose increases by a factor of 4, the noise decreases by a factor of 2.

SEM image noise adds to the actual roughness of the patterns on the wafer, creating a larger biased measured roughness. Typically, a biased roughness is obtained as given by equation 8A.

(8A)

where σ _biased is the roughness measured directly from the SEM image, σ _unbiased is the unbiased roughness (i.e. the actual roughness of the wafer features), and σ _noise is the roughness of the detected edges due to noise in SEM imaging and edge detection. Random errors in position (or linewidth). Equation 8A assumes that noise is statistically independent of the roughness on the feature being measured. Alternatively, more complex noise models may be used, as described further below. Since an unbiased estimate of feature roughness is required, the measured roughness can be corrected by subtracting the estimate of the noise term.

Pixel noise in SEM creates edge detection noise depending on the shape of the expected line scan for the feature. For example, Figure 17A shows a typical line scan (grayscale value versus horizontal position, g(x)) for a line feature on a wafer when there are so many electrons that pixel noise is negligible. The result is, from a statistical standpoint, the "expected" line scan of the line scan signal, i.e. the expected value. By defining a threshold grayscale level, edge positions can be determined. However, noise in the grayscale values results in noise in the detected edge positions. For a given grayscale noise σ _gray , the edge position uncertainty σ _noise will depend on the slope of the line scan at the edge dg/dx. For small level noise,

(8B)

Therefore, the level of edge detection noise is a function of the pixel grayscale noise and the slope of the line scan at the feature edge.

This equation (8B) is strictly valid only for small levels of noise and infinitely small pixel sizes. To explore the impact of larger amounts of noise and non-zero pixel sizes, simulations of SEM images were used. Perfectly smooth lines and spaces (25 nm wide, 50 nm pitch) were used as input to the Analytical Linescan Model to generate synthetic SEM images. The resulting grayscale values (ranging from 0 to 255) for each pixel were then treated as the mean of a normal distribution with a given standard deviation (σ _gray ), and a random grayscale number for each pixel was drawn from this normal distribution. was assigned to Then, these SEM images were processed as experimental SEM images and measured using an inverse line scan model to detect the edge position of each feature. The 1-sigma LER measured from these images is the detected edge position uncertainty due to grayscale pixel noise. Figure 17b shows the 1-sigma uncertainty in edge detection location for these perfectly smooth features in the presence of grayscale noise. In this graph, the edge detection noise for three different is plotted as a function of . Edge detection used the inverse linescan model, and the resulting line edge roughness of the features was considered to be edge detection noise. The results are somewhat non-linear, with higher levels of pixel noise producing much larger edge detection noise. Additionally, smaller X pixel sizes produce lower levels of edge detection noise. In fact, edge detection variance is directly proportional to the X pixel size for low levels of grayscale noise.

Pixel noise is not the only source of edge detection noise. During operation, the electron beam is scanned from left to right using beam steering electronics. Errors in beam steering can place the beam in an incorrect position, creating edge errors. Charging of the sample during electron exposure will deflect the beam to an incorrect location. Some of the charging effects will be systematic, but there will also be random or pseudo-random components that will appear as random fluctuations in the detected edge position.

Several approaches to estimate SEM edge position noise and subtract it have been proposed in the prior art, but these approaches have not proven successful for today's small feature sizes and high levels of SEM image noise. The problem is the lack of edge detection robustness in the presence of high image noise. In particular, when the noise level is high, the edge detection algorithm often cannot find the edge. The solution to this problem is typically to filter the image and smooth out high-frequency noise. For example, if a Gaussian 7x3 filter is applied to an image, for each rectangular region of the image 7 pixels wide and 3 pixels high, the grayscale values for each pixel are multiplied by the Gaussian weight and then averaged together. The result is assigned to the center pixel of the rectangle. Box (mean) filters and median filters can also be used and produce similar results. This smoothing makes edge detection significantly more robust when image noise is high. Figure 17C shows an example of using a simple critical edge detection algorithm with image filtering on the right image and no image filtering on the left image. Without image filtering, edge detection algorithms mostly detect noise within the image and cannot reliably find edges.

The use of image filtering can have a significant impact on the resulting PSD and measured roughness. Figure 18 shows the impact of two different image filters on the PSD obtained from a set of 30 images each containing 12 features. All images were measured using the inverse linescan model for edge detection. The power spectral density was averaged from these 360 coarse features with images preprocessed using a 7x2 or 7x3 Gaussian filter or not filtered at all, as labeled in the figure. As can be seen, the high frequency region is greatly affected by filtering. However, the low-frequency region of the PSD also shows noticeable changes when using a smoothing filter. Filtering in the y-direction smoothes out high-frequency roughness. Filtering in the x-direction lowers the slope of the line scan, which can affect the measured low-frequency roughness. As described below, the use of image filtering makes measurement and subtraction of image noise impossible.

If edge detection can be achieved without image filtering, noise measurement and subtraction can be achieved by comparing the PSD behavior of the noise with the PSD behavior of actual wafer features. Resist features (as well as after-etch features) are expected to have PSD behavior as shown in Figure 19 (and also earlier in Figure 4) as "True PSD". do. Correlations along the length of the feature edge reduce the high frequency roughness so that the roughness becomes very small over very short length scales. On the other hand, SEM image noise can often be assumed to be white noise, so the noise PSD is flat across all frequencies. Other models of SEM image noise are also possible, for example, using linescan-to-linescan correlation to account for noise, as described further below. Therefore, at sufficiently high frequencies, the measured PSD will be dominated by image noise and not by actual feature roughness (the so-called “noise floor”). Given the grid size along the length of the line (Δy), the SEM edge detection white noise affects the PSD according to equation 9 below:

(9)

Therefore, measurement of high-frequency PSD (in the absence of any image filtering) provides a measure of SEM edge detection noise. Figure 19 illustrates this approach for the case of a white SEM noise model. Obviously, this approach to noise subtraction cannot be used for PSDs coming from filtered images, since such filtering removes the high-frequency noise floor (see Figure 18).

Equation 9 assumes a white noise model, where the noise found at any pixel in the image is independent of the noise found at any other pixel. This may not always be the case. For example, noise at each pixel may be somewhat correlated with its nearest neighbors, affecting σ _gray in Equation 8B. Alternatively, the grayscale slope of Equation 8B can be correlated from one row of pixels to the next, possibly caused by the electronic interaction volume shown in Figure 8. If a correlation model is assumed or measured, an appropriate noise expression for the PSD can be used to replace Equation 9, described further below.

19 illustrates one implementation of a noise subtraction process of the disclosed edge detection apparatus and method. In the disclosed edge detection method, the method first detects the location of the edge using ILM (eg, using an inverse line scan method) without using any image filtering. From these detected edges, a biased PSD is obtained, which is the sum of the actual wafer roughness PSD and the SEM noise PSD. Using a model for the SEM image noise (such as a constant white noise PSD), the amount of noise is determined by measuring the noise floor in the high frequency portion of the measured PSD. The actual (unbiased) PSD is obtained by subtracting the noise level from the as-measured (biased) PSD. The key to using the above approach of noise subtraction to obtain an unbiased PSD (and thus the parameters σ _LWR (∞), PSD (0), and ξ) is the ability to robustly detect edges without the use of image filtering. will be. This can be achieved using an inverse line scan model. An inverse line scan model was used to generate the unfiltered PSD data shown in Figure 18.

One example method for subtracting white noise will now be described. First, edges are detected from the SEM image without using any image filtering (eg, using an inverse line scan model). The power spectral density of one or more edges is calculated in the usual way. Because the PSD of a single edge is quite noisy, it is very important to measure many edges and average the PSD. Often hundreds or thousands of edges are measured and their PSDs are averaged. This averaged PSD is called biased PSD. From the average biased PSD, the highest frequency is examined to determine if a flat noise floor is observed. This noise floor is observed whenever the y pixel size is sufficiently smaller than the correlation length of the actual roughness. Typically, a y-pixel size of less than 20% of the correlation length is adequate. Once the noise floor is observed, the average PSD value in the flat region is calculated. This is the noise floor. This number is then subtracted from the biased PSD at all frequencies to produce the unbiased PSD. The biased PSD is the best estimate of the actual PSD of roughness on the wafer.

Other SEM errors can also affect the measurement of roughness PSD. For example, SEM field distortion has little effect on LWR, but can artificially increase the low-frequency PSD for LER and PPR. Background intensity fluctuations in SEM can also cause an increase in the measured low-frequency PSD, including LWR, as well as LER and PPR. If these fluctuations can be measured, they can potentially be subtracted, producing the best possible unbiased estimate of the PSD and its parameters. By averaging the results of many SEM images where the only common mode of measurement is the SEM used, determination of SEM image distortion and background intensity fluctuations can be made.

9. Sensitivity to metrology tool settings

Not all noise in the measured PSD is white noise. White noise occurs when the measurement noise of the edge position from each line scan is completely independent of all other line scans (and especially nearest neighbors). White noise occurs when there is no correlation connecting the error of one line scan with the error of a neighboring line scan. Any small correlation of edge errors along the length of the line will cause “pink noise”, a noise signature that is not completely flat over the entire frequency range.

Settings of the SEM metrology tool can affect the measured roughness of features in the patterned structure. These settings include the magnification and pixel size of the SEM 701. These two parameters can be changed independently by changing the number of pixels in the image (eg, from 512×512 to 2048×2048). Additionally, the number of frames of integration (electron dose) when capturing SEM images can be adjusted. To study the impact of these settings, the number of integration frames can be varied, for example, from 2 to 32, representing a 16X variation in electron dose.

The total electron dose is directly proportional to the number of frames of integration. Therefore, the impact on shot noise and edge detection noise is expected to be proportional to the square root of the number of integrated frames. Figure 20 shows the PSD of a specific resist feature type on a given wafer, measured with different numbers of integrated frames. In this case, the PSD corresponds to 18 nm resist lines and spaces where only the number of integrated frames changes. The SEM conditions used were 500 eV, 49 images per condition, 21 features per image, pixel size = 0.8 square nm, and image size = 1024×1024 pixels. Cases of integration of eight or more frames produce PSDs that exhibit fairly flat high-frequency noise regions. For 2 and 4 integrated frames, the noise area is noticeably sloped. Therefore, the assumption of white SEM noise is only approximately true and becomes a more accurate assumption as the number of integrated frames increases and the noise level decreases. This observation has been proven in other situations: high noise cases are more likely to exhibit non-flat noise floors.

Figure 21 shows biased and unbiased values of 3σ linewidth roughness measured as a function of the number of integrated frames. All conditions were the same as described in Figure 20, and error bars represent 95% confidence interval estimates. The biased roughness varies from 8.83 nm in two integrated frames to 5.68 nm in eight frames and 3.98 nm in 32 frames. Meanwhile, the unbiased roughness is fairly stable after six integrations, varying from 5.25 nm in two integrated frames to 3.25 nm in eight frames and 3.11 nm in 32 frames. The biased roughness is 43% higher with 8 frames compared to 32, but the unbiased roughness is only 4% higher with 8 frames compared to 32. Because the assumption of white SEM noise is not very accurate in the two and four frames being integrated, noise subtraction of unbiased measurements using a white noise model is not completely successful in these very small numbers of frames of integration. The correlated noise model can produce better noise subtraction, especially for small numbers of integrated frames, as explained more fully below. Although the results shown are for LWR, similar results are obtained for measurements of line edge roughness (LER) and pattern placement roughness (PPR).

One possible cause of correlations in edge noise would be correlations in pixel noise. To test this possibility, isolated edges were measured by CD-SEM. The edge allows the SEM to perform its imaging function in a typical manner, but at distances to the left or right of the edge the field is flat and featureless. The only variation in pixel grayscale values in this region comes from image noise. Then, the correlation coefficient between neighboring pixels can be calculated. Performing these calculations, the average correlation between neighboring pixels in the x-direction was 0.12, but the average correlation in the y-direction was only 0.01, essentially zero. These correlation coefficients were determined for edges measured in the integration of 2 to 32 frames. There was little variation in pixel-to-pixel correlation as a function of the number of frames of integration. Therefore, correlated pixel noise is not responsible for the pink noise observed in low integration frames. However, it is possible that the line scan slope in Equation 8B is responsible for the noise correlations.

A possible source of noise correlation in the line scan slope comes from the interaction of the beam with the sample. Electrons impinging on a sample undergo a number of processes that depend on the energy of the electrons and the material properties of the sample. Electrons scatter atoms of the sample material, releasing energy, changing direction, and often creating a cascade of secondary electrons by ionizing the sample atoms. Sometimes electrons bounce back from the atomic nucleus and escape the sample (called backscattered electrons). Some of the lower energy secondary electrons may also escape out of the sample (often through the edges of the feature, see FIGS. 8A and 8B) (see also). The way an SEM forms an image is by detecting for each beam position the number of secondary and/or backscattered electrons escaping the sample.

When forming an image using an SEM, a small spot of electrons resides at a specific point on the sample (i.e., pixel), while the number of secondary electrons that escape is counted by a secondary electron detector. As in Figure 8A, when the spot is a long distance from the feature edge, the number of secondary electrons 805 detected is small (and the pixel is dark). When the spot is near a feature edge, as in Figure 8B, secondary electrons 805 from the interaction volume easily escape from the feature edge producing bright pixels.

The interaction volume of electrons can be from one to several tens of nanometers in diameter depending on beam voltage and sample material properties. This interaction volume means that electrons striking a spot on the sample will be affected by the sample geometry over a range determined by the interaction volume. Therefore, the slope of the line scan in one row of pixels will not be independent of the slope of the line scan in neighboring pixels, whenever the interaction volume radius is greater than the y pixel size. This dependence can result in correlations in noise, with the noise correlation length being influenced by the electron beam interaction volume.

10. Spike detection and removal in power spectral density

In addition to noise interfering with the signal in typical images of rough features, there may be other errors in the images that have a very different frequency behavior compared to white or pink noise and compared to the roughness being measured. Some of these errors produce large but narrow spikes in the PSD. Figure 25A shows an example of high frequency “spikes” found intermittently in datasets. One cause for such spikes could be electrical interference in the imaging tool's scanning electronics. If the interference is in a range of frequencies that allow one or more interfering events within the entire scan of the image, this interference can result in a slight but regular "jitter" of the scanning beam position. For very precise scanning, even sub-nanometer jitter can result in one or more large spikes in the measured PSD. Depending on the mechanism, these interfering spikes may be present in the line edge roughness (LER) and pattern placement roughness (PPR), but not in the linewidth roughness (LWR) PSD. Alternatively, interference may cause spikes at the same frequencies in all three PSDs.

For example, electrical interference at frequencies of 50 Hz or 60 Hz can cause noticeable spikes in the measured PSD when the measurement tool captures images at the standard “TV” scan rate or small multiples of this rate. Additionally, electrical interference at normal audio frequencies can cause visible spikes at higher PSD frequencies in typical measurement tool images.

The presence of spikes in a PSD can be undesirable for a number of reasons depending on their amount, their amplitude, and their frequency. In the case of high frequency spikes as shown in Figure 25A, the spikes can affect the noise removal process described above, resulting in an overestimation of the amount of white or pink noise in the image.

PSD spikes can be caused by phenomena other than electrical interference within the imaging tool. The object being measured may include periodic or semi-periodic structures other than the coarse features to be measured. For example, a set of vertically oriented rough features of an object may be on top of a periodic set of horizontal features, resulting in a topography below the rough features that are slightly visible in the image. This underlying topography can result in a mid-frequency spike for the PSD (higher frequency harmonics are also possible). Figure 26 shows an example of this phenomenon.

Another phenomenon that can cause spikes in PSD would be the presence of grains in a small size range within the material of the feature on the object being measured. Similar sized particles packed tightly together can create a nearly periodic appearance that results in spikes in the measured PSD.

Roughness measurements can also be performed on captured images of photomask features, where photomasks are used in a lithography process. Photomasks are typically manufactured using a direct-write lithography tool with limitations such as a non-zero address grid and rectangular shots to construct the image. For some features, such as lines oriented at 45 degrees to the direction of the recording grid of the tool used to print the photomask, the result will be small, regularly spaced jogs along the edges of the photomask feature. These jogs will generate spikes (or main spikes and harmonic spikes) in the PSD of the measured photomask roughness.

Spikes such as those found in Figure 26 can be very detrimental to the measurement of roughness parameters from biased or unbiased PSD. Figure 27A shows how a PSD with spikes, including modeling parameters such as PSD(0), correlation length, and roughness exponent, can change the model fit to an unbiased PSD. In contrast, Figure 27b shows how a PSD with spikes removed can affect a model fitted to an unbiased PSD, including modeling parameters such as PSD(0), correlation length, and roughness exponent.

For these and other reasons, it is desirable to remove spikes from the PSD when the cause of these spikes is thought to come from a different mechanism than the mechanism that causes the roughness of the features being measured. That is, it is important to separate PSD artifacts caused by one mechanism (such as spikes caused by electrical interference) from PSD artifacts caused by other mechanisms (such as stochastic effects that cause roughness). desirable. This can be done very similarly to noise removal described above, by recognizing the different frequency signatures of different mechanisms.

As mentioned above, white noise (or pink noise) is very different from the frequency signature of the actual roughness (which is flat or nearly flat at high frequencies) because the frequency signature of the noise (which is flat or nearly flat at high frequencies) is very different from the frequency signature of the actual roughness (which is a decreasing power-law at high frequencies). The (unbiased) roughness can be separated from the PSD. Likewise, these so-called spikes in the PSD have very different frequency signatures than the frequency signature of the feature roughness itself. In particular, so-called spikes have high amplitude over a very narrow frequency range.

The procedure for detecting and removing spikes is now described. First, the definition of a “spike” is that it rises and falls over a frequency range less than a threshold (the “threshold range”) and has a height greater than the threshold (the “threshold height”). It can be established that the frequency response has

Next, a baseline can be established as being the best estimate of the spike-free PSD. For example, the threshold range for spike detection can be set to three frequency increments in PSD data (typically sampled at constant frequency increments). Other critical ranges are also possible. The baseline can be determined by seamlessly connecting the separated PSD values by the critical range plus one increment (e.g., using a straight line on a linear or logarithmic scale, or using a model for expected PSD behavior). This baseline is then subtracted from the actual PSD data within this threshold range to arrive at an estimate of the non-baseline PSD behavior within this frequency range. A spike was identified if the non-baseline PSD behavior rose to a value greater than the threshold height (expressed in absolute terms or as a multiple of the baseline PSD value). To remove spikes, the calculated baseline behavior can be used to replace the actual PSD values within the threshold range. Searching for spikes can cover the entire PSD frequency range if desired.

The threshold range can be selected in such a way that it detects (and possibly eliminates) only spikes that result from a particular type of mechanism. For example, interference at exactly a single frequency will most likely cause a spike in a PSD that spans up to two frequency increments (since the spike will not be at a frequency that exactly matches the sampled frequencies of the PSD). . A wide threshold range of 2 to 3 frequency increments will be effective in detecting these “single frequency” interference events. A wider threshold range will detect other wider bands of interference events.

The threshold height can also be adjusted based on the mechanisms desired to be detected. However, the minimum threshold height is also a function of the overall noise in the PSD. Because PSD, by definition, measures randomness in random samples, PSD measurements are inherently noisy. It is well known that the PSD of a single measured feature has a statistical uncertainty of 100% (1-sigma). That is, the statistical uncertainty of any given PSD value at any given frequency is 100% for the measurement of a single feature. For that reason, many features are typically measured together and averaged, so that the uncertainty in the PSD can be reduced by 1 over the square root of the number of features being measured.

However, for any given number of features measured and averaged, the PSD will have statistical uncertainty inherent to the sample size. The threshold height for spike detection should be chosen to be significantly higher than the inherent noise level of the PSD. Otherwise, detection of spikes will frequently be triggered by noise in the PSD data and not by physical spikes. Alternatively, the threshold height may be selected to be a multiple of the measured or calculated PSD noise (e.g., 5X).

25A shows several PSDs (linewidth roughness (LWR PSD 2502), line edge roughness (LER PSD 2504), and pattern placement roughness (PPR) showing several high frequency spikes (spike artifacts 2507). PSD 2506. Figure 25B shows the same PSDs (e.g., LWR PSD 2502 as LWR PSD 2508, LER PSD 2510) from which spikes were removed using the procedure outlined in the previous paragraphs. ) as LER PSD 2504, and PPR PSD 2506 as PPR PSD 2512. For this removal, the threshold range was set to three frequency increments, and the threshold height was set to three times the baseline PSD value. Effective removal of spikes was achieved using these settings.

Figures 27a and 27b show another case of spike rejection, this time for mid-frequency spikes. The left graph in Figure 27A shows the PSD (biased and unbiased) before spike removal. The presence of spikes has a deleterious effect on the modeling of PSD and the extraction of PSD measurements. The graph on the right, Figure 27b, shows the same PSD with spikes removed using the procedure outlined in the previous paragraph. For this removal, the threshold range was set to three frequency increments and the threshold height was set to three times the baseline PSD value. Effective removal of spikes was achieved using these settings. The resulting PSD modeling and PSD measurements more accurately reflect feature roughness PSD behavior excluding the mechanism that generated the spikes.

An alternative procedure for removing spikes will now be described. Spikes can be removed from the PSD by passing the PSD through a low-pass filter. Using well-known techniques, the PSD can be Fourier transformed, multiplied by a low-pass frequency filter, and then inverse Fourier transformed. The cutoff frequency of the low-pass filter can be set to only filter out spikes narrower than a set threshold. Other approaches to low-pass filtering known in the art may also be applied.

Other methods for detecting and removing spikes will be known to those skilled in the art based on the different frequency characteristics of the spikes compared to the more slowly changing true roughness PSD.

7, an information processing system (IHS 750) may be modified to include detection and/or removal of spikes using one of the example methods described herein. Information about each detected spike, such as its center frequency, amplitude, area, and/or width, may be recorded and output to output device 770. This information can be useful in identifying the root cause of spike formation and thus aid in the process of reducing or eliminating that root cause mechanism.

11. Detection and measurement of PSD bumps

Other phenomena can cause PSD behavior to appear as “bumps” in the PSD, which otherwise have the typical shape shown in Figure 3. These bumps typically occur at relatively low frequencies. These bumps are distinguished from spikes by covering a relatively wide range of frequencies, in contrast to the narrow frequency limits of spikes. Figures 28a and 28b show two examples of this so-called bump behavior in PSD, labeled as bump type I and bump type II.

Bump type I (Figure 28a) is a large rise in low frequency PSD behavior beyond what is generally considered to be a flat low frequency region characterized by PSD(0). Several mechanisms can cause these bumps, such as the presence of photomask roughness transferred to the wafer during the photolithography step. Uncompensated field distortions in the imaging tool used to capture the images being measured can also cause this type of bump. Other mechanisms are also possible.

Bump type II (Figure 28b) occurs at low-to-mid frequencies such that the PSD behavior at frequencies above and below the bump follows the expected behavior (e.g., as seen in Figure 3). do. When this type of PSD bump is found in the line edge roughness PSD but not in the linewidth roughness PSD, the effect is sometimes referred to as "wiggle" because it is noticeable as a wiggle in the features of the image. Such waviness may result, for example, from stress or tension in the film used to fabricate the feature. Photolithography and subtractive etching of the film to form features can relieve stress and allow the remaining film to be relieved of tortuosity. Other mechanisms causing wiggle are also possible.

Like white noise and spikes, bumps in the PSD are thought to occur through mechanisms that are separate from the stochastic mechanisms that generate the rest of the PSD. Therefore, it is desirable to separate the effects of the bump from the rest of the PSD. Similar procedures to spike detection and removal can be used for bump detection and removal. However, this approach becomes problematic when the width of the bump is large due to the difficulty of defining the baseline PSD behavior over a large frequency range. The wide frequency range of bumps means that bumps and spikes can be distinguished, but it also means that different procedures are likely needed to detect and measure the bumps.

A separate technique for bump detection, measurement and removal involves the use of a model for the bump. Like white noise and pink noise, the bump model is added directly to the typical PSD of feature roughness. Therefore, the bump model can be fit to the PSD simultaneously with a typical PSD model that does not include bump behavior.

A useful form for the bump model is given in equation 10 below:

(10)

Here, A is the amplitude of the bump, f _c is the center frequency of the bump, and σ _w is the width of the bump. For a Type I bump (Figure 28a), the center frequency may be zero. Other models may also be used. Alternative parameterizations of the model, such as area and center frequency of the bump, may also be used.

For example, the area of the bump above the baseline PSD, as determined from a best fit model, is a useful measure of the magnitude of the phenomenon that gave rise to the bump. For example, for serpentine (bump type II example, Figure 28b), the area represents the variance of the serpentine added to the variance caused by stochastic roughness. That is, this approach for bump detection and measurement allows the total variance of the features to be separated into wiggle variance plus stochastic roughness variance.

Referring to Figure 7, IHS 750 may be modified to include detection and/or removal of bumps using one of the example methods described herein. Information about each detected bump, such as center frequency, amplitude, area, and/or width, may be recorded and output to output device 770. This information can be useful in identifying the root cause of bump formation and thus aid in the process of reducing or eliminating that root cause mechanism. By subtracting the bump behavior from the total PSD, the remaining PSD can be characterized (e.g., using parameters such as PSD(0), correlation length, and roughness exponent) such that this remaining PSD is the bump mechanism. It more accurately reflects the mechanisms that generate PSD, except for .

Referring now to FIG. 30, an example method 3000 for detecting unwanted spikes in and removing spikes from a PSD dataset is illustrated. Method 3000 begins (block 3002), using an imaging device to generate a set of one or more images, each image of the set comprising an instance of a feature within a respective pattern structure, and each image containing noise Includes measured line scan information corresponding to a pattern structure comprising (block 3004). Next, the method proceeds to detect edges of features within the pattern structure of each image of the set without filtering the images (block 3006), and generates a power spectrum representing feature geometric information corresponding to the edge detection measurements of the set of images. Create a density (PSD) dataset (block 3008). If desired, an unbiased PSD data set can be generated from a biased PSD data set by subtracting the SEM noise. Next, the method defines a threshold range and a threshold height (block 3010) and creates a baseline for the portion of the PSD dataset by seamlessly connecting the first PSD value of the portion to the second PSD value. and wherein the first PSD value and the second PSD value are separated by a threshold range (block 3012) and determine that the difference between the third PSD value of the portion of the PSD dataset and the baseline is greater than the threshold height (block 3014 ), replace the portion of the PSD dataset with the baseline for the portion of the PSD dataset (block 3016). The method then ends (block 3018).

Referring now to Figure 31, an example method 3100 for modeling bumps in a PSD dataset is illustrated. Method 3100 begins (block 3102), using an imaging device to generate a set of one or more images, each image of the set comprising an instance of a feature within a respective pattern structure, and each image containing noise Includes measured line scan information corresponding to a pattern structure comprising (block 3104). Next, the method proceeds to detect the edges of features within the pattern structure of each image of the set without filtering the images (block 3106) and generates a biased graph representing feature geometry information corresponding to the edge detection measurements of the set of images. Generate a power spectral density (PSD) dataset (block 3108). If desired, an unbiased PSD data set can be generated from the biased PSD data set by subtracting the SEM noise. A first bump is evaluated in the PSD dataset to create a bump model (block 3110); Fit a typical PSD model and a bump model to the PSD dataset to generate a best fit model (block 3112). The method then ends (block 3114).

The flow diagrams of FIGS. 30 and 31 illustrate specific steps that may be performed by SEM 701 and IHS 750 and its included processor 755 and storage 760, both of which are described in detail herein. - steps that may be performed using the edge detection system 700 shown in FIG. 7, including certain other steps that may be performed by. The instructions, when executed by a processor, cause the processor to: It may be stored in storage 760 to perform the methods described by the flowcharts of FIGS. 30 and 31.

12. Impact of pixel size and scale

With respect to the pixel size and magnification employed by the SEM 701, Figures 22a and 22b show the bias and ratio for a pattern of 16 nm lines and spaces for different magnifications and pixel sizes, assuming a white noise model. The biased power spectral densities (PSDs) are shown individually. For a given number of integrated frames, varying the pixel size changes the electron dose per unit wafer area and the noise in the SEM image. Under this range of conditions, the biased LWR changed by 0.63 nm (14%), while the unbiased LWR changed by only 0.07 nm (2%). Unbiased LWR is essentially unaffected by these measurement tool settings. Similar results were obtained for measurements of LER and PPR.

Figures 22A and 22B show power spectral density as a function of pixel size and magnification. More specifically, Figure 22A shows the biased LWR PSD and Figure 22B shows the unbiased LWR PSD after noise is measured and subtracted. SEM conditions for these results used a landing energy of 500 eV, three images per condition, and 16 nm resist lines and spaces.

Table 3 below shows the measured PSD parameters for the PSDs shown in FIGS. 22A and 22B.

Table 3

Biased and unbiased 3σ LWR (nm) measurements as a function of pixel size and magnification.

It was found that the difference between biased and unbiased LWR is not constant, but varies depending on metrology tool settings, feature size, and process. Likewise, the ratio between biased and unbiased LWR varies depending on metrology tool settings, feature size, and process. Table 4 below shows the difference and ratio of biased to unbiased LWR for various conditions. For these conditions, the ratio of biased to unbiased LWR varies from 1.09 to 1.66. The difference between biased and unbiased LWR varies from 0.32 nm to 2.19 nm in this particular example.

Table 4

Relationship between biased and unbiased LWR for various processes.

13. Implementing edge detection

FIG. 23 is a flow chart illustrating a representative overall process flow employed by the disclosed SEM edge detection system to detect edges of a pattern structure. For discussion purposes, the process described in the flow chart of Figure 23 is applied to sample 2400 of Figure 24A. Sample 2400 is a pattern structure, which may also be referred to as pattern structure 2400. The flow diagram of FIG. 23 includes steps performed by the inverse line scan model metrology tool 765 to determine the edges of the pattern structure.

The process flow begins at start block 2300 in Figure 23. As shown in Figure 7, IHS 750 is coupled to SEM 701 and receives SEM line scan image information from SEM 701. IHS 750 includes a processor 755 and storage 760 coupled thereto. Storage 760 may include volatile system memory and non-volatile persistent memory, such as hard drives, solid state storage devices (SSDs), etc., that permanently store applications and other information. Storage 760 stores an inverse linescan model (ILM) metrology tool 765 disclosed herein and illustrated by the flow diagram in FIG. 23. SEM 701 includes a controller (not shown) that directs IHS 750 to perform image acquisition on pattern structure 800 and provides line scan information from SEM 701 to IHS 750.

According to block 2305, SEM 701 sends an SEM image of pattern structure 800 to IHS 750, and in response, IHS 750 stores the SEM image in system memory in storage 760. Loading. IHS 750 preprocesses the pattern structure image from SEM 701 according to block 2310. For example, such preprocessing of loaded SEM images may include adjusting grayscale values and subtracting background tilt of intensity levels. Optionally, according to block 2315, IHS 750 may perform filtering of the loaded image, although this is generally undesirable.

For pattern structures, such as the vertical lines and spaces seen in pattern structure 2400 of FIG. 24A, inverse line scan model metrology tool 765 perpendicularly about the axis of symmetry to generate an average line scan according to block 2320. average out The average line scan can be a grayscale value as a function of horizontal position and all vertical pixels are averaged together. This produces a line scan that is more representative of the physical process by averaging out most of the SEM noise contained in the SEM image, producing a noise-free line scan. Figure 24b shows a single line scan at one Y-pixel location. Figure 24C shows an averaged line scan produced by averaging across all Y-pixels.

The example shown here is for vertical lines and spaces, but any pattern with an axis of symmetry can be so processed to produce an average line scan. For example, long lines, long spaces, or long isolated edges may be so processed whenever the length of the line is sufficient to allow proper averaging. Contact holes or pillars with circular or elliptical symmetry can also be radially averaged to produce an average line scan.

Following block 2325, tool 765 calibrates the inverse line scan model against the average line scan obtained in the manner described above. Note that the line scan model includes two kinds of parameters: 1) parameters that depend on the properties and materials of the SEM, and 2) parameters that depend on the geometry of features on the sample. Tool 765 can calibrate all of these parameters. Tool 765 follows block 2325 to find the best fit of the model to the average line scan of FIG. 24C. Then, the values of the model's best fitting parameters are the calibrated values.

The calibrated model is applied to the single line scan shown in Figure 24b. The best fit of the model is found for the single line scan of Figure 24b, but in this case, the inverse line scan model metrology tool 765 fixes both the parameters related to the materials and the SEM imaging tool. In this scenario, tool 765 only changes parameters related to the geometry of the features of the pattern structure to find the best fit of the calibrated model for a single line scan.

In a simplified scenario, the only parameters changed at block 2330 would be the positions of the features' edges. In one implementation, the vertical dimension of the feature is assumed to represent a predetermined thickness and only the edge positions of the feature change. Next, the calibrated inverse line scan model is fit to every single horizontal cross section through the 2D image of the feature, according to block 2330. Take the top horizontal row of pixels, then take the next pixel row one pixel down, then take the next pixel row one pixel down. An example of one such single line scan is shown in Figure 24B. The resulting best fitting edge positions are the detected edges.

After the edges of the feature have been detected in the manner described above, tool 765 may detect that the sample was slightly rotated during image acquisition, resulting in parallel tilted lines (i.e., lines that are not completely vertical). . This tilting or rotation can contribute to the inaccuracy of the detected edges by changing the average line scan and thus the calibrated ILM. Image rotation can be detected by fitting all edges of the image to a set of parallel lines and determining their tilt relative to the vertical. If the tilt is sufficiently different from the vertical case, the rotation should be removed. One possible criterion would be to compare the pixel location of the best fitting line at the top of the image with the pixel location of the best fitting line at the bottom of the image. If these pixel positions differ by some threshold, such as two pixels, the image rotation is considered large enough to require its removal.

If such tilt/rotation is detected according to block 2335, the previous calibration is considered a first pass calibration and the calibration is repeated. More specifically, when such a tilt/rotation is detected, according to block 2345, the rotation is subtracted by shifting some rows of pixels to vertically align the edges and calculating a new average line scan. The calibration of the model is then repeated according to blocks 2350 and 2325. According to block 2330, another fitting is also performed. Ultimately, tool 765 outputs geometric feature information (e.g., edge positions) that describes the geometry of the feature that corresponds to the line scan image information provided to tool 765.

Like image rotation, the roughness of the features themselves contributes inaccuracy to the ILM's calibration. Optionally, after first pass edge detection, each row of pixels can also be shifted to subtract image rotation as well as subtract feature roughness. The end result after shifting each row of pixels is a vertical edge where the edge position varies by less than 1 pixel from a perfectly vertical line. These shifted rows of pixels can then be vertically averaged to produce a more accurate average line scan for use in ILM calibration.

In actual implementations, the IHS 750 may include an interface 757 coupled between the processor 755 and an output device 770, such as a display, printer, or other user interface, allowing the user to perform inverse line scan model metrology. Feature edges determined by tool 765 can be observed. Interface 757 may be a graphical interface, printer interface, network interface, or other hardware interface suitable for a particular type of output device 770.

Although the above-described embodiments refer to the measurement of structures found on semiconductor wafers, used in the manufacture of semiconductor devices, the invention is not limited to these applications. The present disclosure may be useful for measuring the roughness of feature edges found in flat panel displays, microelectromechanical systems, microfluidic systems, optical waveguides, photonic devices, and other electronic, optical, or mechanical devices. Additionally, the present invention can be used to measure feature edge characteristics of naturally occurring structures, such as crystals or minerals, or manmade structures, such as nanoparticles or other nanostructures. Moreover, the present invention can also be used to measure feature edge properties of biological samples.

Although the above-described embodiments refer to measurements using a scanning electron microscope, the invention is not limited to that imaging tool. Other imaging tools may also be used, such as optical microscopy, stimulated emission and depletion (STED) microscopy, x-ray microscopy, transmission electron microscope (TEM), focused ion beam microscopy, and helium ion microscopy. Other types of microscopes may also be used, such as scanning probe microscopes (atomic force microscopes) (AFMs) and scanning near-field optical microscopes (SNOMs).

14. Prior art methods for determining process window

In lithography, two key variables are frequently adjusted to maintain control of the final printed features: exposure dose (also referred to as the product of exposure energy, light intensity, and exposure time) and focus. (Often described as the location of the best focal plane of the projected image relative to the top surface of the photoresist coated on the wafer). Both of these process variables can be altered using the lithographic projection imaging tool (e.g., a stepper, scanner, or step-and-scan tool) used to print the features. there is. Other lithographic techniques such as direct-write patterning, proximity printing, and electron-beam lithography can also adjust exposure dose and focus or their equivalents.

To obtain the best lithography results, it is desirable to set the exposure dose and focus of the lithography tool to optimal values, sometimes called best dose and best focus. The meaning of “best” may vary depending on the goals of the lithography process. One common goal would be to ensure that the mean feature size of a particular printed pattern on a wafer or set of wafers matches a target value for that feature size, within some tolerance. Another goal would be to minimize the variation in feature size over some set of features. Another goal would be to improve the pattern fidelity of printed patterns. Another goal would be to minimize the sensitivity of printed feature size or fidelity to process variations over some set of features.

The set of features inspected can be features printed at different points in the imaging field (or different points within the scanner slit area for scanning lithography), different points across the chip being fabricated, different points across the wafer, and different points across different wafers or different lots of wafers. The set of features inspected may also include features printed at different times. The set of inspected features may also include different feature types, features with different proximity to other features, and features with different target feature sizes.

In many lithography processes, feature size varies monotonically with exposure dose. For example, for small ranges of doses, feature size may vary approximately linearly with dose. For some processes, feature size may be approximately proportional to 1 to exposure dose. In the case of focus, feature size often varies roughly in two dimensions with focus. Additionally, focus and dose typically interact to cause changes in printed patterns. The effect of the dose on the feature depends on the focus setting of the exposure tool, and the effect of the focus on the feature depends on the exposure dose setting of the exposure tool.

Characterizing the effects of focus and exposure dose on printed patterns is typically accomplished using a focus-exposure matrix (FEM). Descriptions of FEM can be found in prior art, for example in the textbook Chris A. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication, John Wiley & Sons, (London : 2007).

In general, depth of focus (DOF) can be thought of as the range of focus errors that a process can tolerate and still provide acceptable lithography results. A change in focus results in two major changes to the final lithographic result: the photoresist profile (including its dimensions) changes, and the sensitivity of the process to other processing errors changes. Typically, a photoresist profile has parameters such as linewidth (or critical dimension, CD), edge placement error, sidewall angle of the profile, final height of the feature, line edge or linewidth roughness, and other metrics known in the art. It is explained using . The focus and variation of these parameters can be determined for any given set of conditions using a variety of measurement methods. For example, CD can be measured using scanning electron microscopy.

Additionally, when the image is out of focus, the process depends on other factors such as exposure dose, post-exposure bake time and temperature, development time, developer concentration and temperature, underlying filmstack properties, etc. Becomes more sensitive to changes in processing variables. Exposure doses are generally selected to exhibit these different process responses. The effect of focus on the sensitivity of patterning to exposure dose can be characterized by varying both exposure dose and focus, respectively, over some set range and measuring properties of the printed patterns such as CD, linewidth roughness, etc.

One way to display the combined influence of focus and exposure dose on CD is to use what is called a Bossung plot, shown in Figure 39A. What is not shown in Figure 39A is the experimental error (or experimental uncertainty) that is always present in experimental data. Experimental error/uncertainty (often expressed as a multiple of the standard error of the measured value, such as twice the standard error) can be 1 to a few percent of CD at best focus and exposure, but the error can be much higher when out of focus. You can. To better analyze focus-exposure CD data, one common approach in the art is to use reasonable empirical equations to reduce data noise and remove flyers (also referred to as outliers). Fitting the data. A drawback of this approach is that the choice of empirical representation adds a level of arbitrariness to the resulting analysis.

One output as a function of two inputs can be plotted in several different ways. For example, Bossung curves can also be plotted as exposure latitude curves (linewidth versus exposure dose) for different focus settings. Another very useful way to plot this two-dimensional data set is a contour plot, which is a contour of constant linewidth versus focus and exposure (Figure 39b). For noisy experimental data, it is also common to plot a smoothed version of the data in such a contour plot. For example, as mentioned above, you could use a curve fitting function to smooth the data and then plot contours based on the curve fitting instead of the contours of the raw data.

The contour plot form of data visualization is particularly useful for establishing exposure and focus limits that will ensure that the final image meets specific specifications. Rather than plotting all contours of a given CD, one can plot only the two CDs that correspond to the outer limits of acceptance - the minimum CD and the maximum CD allowed based on the CD specification. Because of the nature of contour plots, other variables can also be plotted on the same graph. Figure 39C shows an example where the contour of the CD (nominal +/-10%), 80 degree sidewall angle, and 10% resist loss are all plotted on the same graph. The result is a process window - a region of focus and exposure that keeps the final printed pattern within all applied specifications.

Many different measurement results can be combined into a process window. For feature sizes and lithography processes where the stochastic variation is significant, stochastically related results can be plotted on process window contours. For example, linewidth roughness can be plotted using a single specification - maximum allowable LWR. Defectivity can also be used as a probabilistic metric.

The focus-exposure process window is useful because it shows how exposure and focus work together to affect linewidth and/or other metrics used to judge printed results. The process window can be thought of as the process capability of how the process responds to changes in focus and exposure. Analysis of error sources for focus and exposure in a given process will provide process requirements. If the process capacity exceeds the process requirements, the yield of devices manufactured will be high. However, if wider process requirements do not fit within process capabilities, yield or device performance may be compromised.

It is often useful to evaluate the maximum focus range and exposure that can fit within the process window (i.e., maximum process requirements). A simple way to investigate this question is to graphically represent errors in focus and exposure as rectangles on a plot identical to the process window. The width of the rectangle represents the built-in focus errors of the processes, and the height represents the built-in dose errors. The problem then becomes one of finding the largest rectangle that fits within the process window.

There may be many possible rectangles of different widths and heights that are 'maximal', i.e. they cannot be made larger in either direction without extending beyond the process window (Figure 40a). Each maximum rectangle represents one possible trade-off between tolerance for focus error and tolerance for exposure error. When exposure errors are minimized, a larger DOF can be obtained. Likewise, exposure latitude can be improved if the focus error is small. The result is a trade-off between exposure latitude and DOF.

If all focus and exposure errors were systematic, the graphical (or geometric) representation of those errors would be rectangular. The width and height will indicate the total extent of the individual error. However, if the errors are randomly distributed, a probability distribution function will be needed to account for them. Random errors in exposure and focus are caused by the sum of many small error sources, so the central limit theorem makes it common to assume that the overall probability distributions for focus and dose will be approximately Gaussian (normal). am.

In order to graphically represent errors in focus and exposure, a surface with a certain probability of occurrence must be described. Any errors in focus and exposure within the surface will have a greater probability of occurrence than the established cutoff. For a Bi-Gaussian distribution (a Gaussian distribution for two independent variables), the surface of constant probability is an ellipse (Figure 40b).

Using a rectangle for systematic errors or an ellipse for random errors, the size of errors that can be tolerated for a given process window can be estimated using this geometric approach. Taking a rectangle as an example, we can find the largest rectangle that will fit within the process window. Figure 41 shows an analysis of the process window in which all maximum squares are determined and their height (exposure latitude) versus their width (depth of focus) is plotted. Likewise, assuming Gaussian errors in focus and exposure, any maximum ellipse that fits within the process window can be determined. The horizontal width of the ellipse may represent a 6-sigma error in focus (plus and minus 3-sigma variation for best focus), while the vertical height of the ellipse will give a 6-sigma error in exposure. The height versus width of all maximum ellipses was plotted to obtain the second curve of exposure latitude versus DOF in Figure 41.

The definition of depth of focus also naturally leads to the determination of best focus and best exposure. The DOF value corresponding to a point on the exposure latitude versus DOF curve corresponds to the single largest rectangle or ellipse that fits within the process window. The center of this rectangle or ellipse will then correspond to the best focus and exposure for this desired operating point. Knowing the optimal focus and dose values is essential to be able to utilize the entire process window. If the process focus and dose settings deviate from this optimum, the range of allowable focus and dose errors will be reduced accordingly.

Although all of the above results describe the focus and exposure response of a single critical feature, in practice multiple mask features must be printed simultaneously. For example, features of different nominal sizes and pitches or different proximity to other features may exist within the same design. In such cases, the overall process window will be a superposition of the process windows for each feature size or type under consideration. Focus and dose settings within the overlapping process window will allow each feature type and/or size to meet specifications. A more important performance measure than the DOF of each individual feature is the overlapping DOF of multiple critical features.

Just as multiple profile metrics overlap to form one overlapping process window in Figure 39C, process windows from different features can overlap to determine the DOF for printing multiple of these features simultaneously. Figure 42 shows an example of a line/space pattern of two different pitches. Ideally, the process windows for all critical feature sizes and pitches overlap.

Systematic variations in patterning as a function of field position can also be described by overlapping process windows. If the same feature was printed at different points within the lithography tool image field (typically the center of the field and four corners are sufficient), the process windows can be overlapped to create a usable process window for that feature. The resulting depth of focus is sometimes called the usable depth of focus (UDOF).

FEM analysis can also be used in other ways. One possible output of process window analysis is isofocal bias. Typically, there is one exposure dose that causes printed features to have minimal sensitivity to changes in focus. This dose is called isofocal dose. At this dose, the printed CD is generally different from the target or desired CD. This difference is called isofocal bias. The process window can be used to evaluate isofocal bias.

The prior art geometric methods for process window determination and analysis as described above have several problems. Firstly, measurement errors or measurement uncertainty are not taken into account. The contour lines shown in Figure 39C or Figure 40A show sharp boundaries that depict the area inside the process window as separate from the area outside the process window. This sharp boundary creates a hard cut-off between areas considered "good" (inside the process window) and areas considered "bad" (outside the process window). In reality, the boundary between good and bad is vague due to measurement uncertainty.

A second problem with prior art geometric methods is the need for curve-fitting or smoothing of the measurement data before constructing the process window. Just one anomalous data point near the process window edge can produce large variations in contour position (Figure 43). The result can be very significant differences in DOF determined from this process window, and/or large differences in best focus and exposure.

Using smoothing or curve fitting of the raw data adds randomness to the data analysis. There is no a priori exact amount of smoothing or correct fitting function to use, and different choices produce different results.

The third issue is the choice between using rectangles, ellipses, or some other geometric shape to represent the probabilities of focus and exposure dose errors relative to the best focus and dose values. As shown in Figure 41, different choices produce different results. Additionally, using geometric shapes to represent these probabilities and then finding the maximum shape that fits within the process window means that only a few points of the process window shape affect the result. As shown in Figure 40B, the inscribed ovals and squares touch the process window at only a few points. This makes the size of such inscribed features particularly sensitive to metrology errors that may affect only a few points of the contour-based process window.

All of the above issues are highlighted when overlapping pattern size or more metrics of fidelity or roughness, or when overlapping process windows from different features.

Problems with geometric process window analysis result in inaccurate and/or imprecise measurements of exposure latitude and depth of focus. The result of inaccurate or imprecise measurements of exposure latitude or depth of focus is poor optimization of the patterning process. Often material selections (different photoresists, different underlying layers beneath the photoresist, etc.) are made based on (or considering) which material provides the greatest depth of focus, for example. Additionally, process tool settings, such as the size and shape of the illumination used by the light projection tool, take into account the resulting depth of focus.

Optical Proximity Correction (OPC) is used to correct the chip design represented on the photomask to improve the fidelity of the final printed pattern. Sometimes these OPCs consider the impact of these photomask modifications on the process window. Therefore, inaccurate or imprecise determination or analysis of the process window can have a detrimental effect on the results of OPC.

Another consequence of the problems of geometric process window analysis is inaccurate and/or imprecise measurement of best focus and best exposure. The output of best focus and best exposure from process window analysis can be used to control lithography exposure tools during manufacturing. The measured values of best focus and dose are sent to the lithography tool and used to adjust the dose and focus settings of that tool for subsequent use in manufacturing product wafers. Inaccurate and/or imprecise measurements of best focus and best exposure can then ultimately negatively impact device yield or device performance.

Additionally, the exposure settings of a lithography tool can affect the throughput of that tool. Generally, lower exposure doses produce higher throughput of the exposure tool. For this reason, it may be desirable to lower the exposure dose as much as possible while still maintaining appropriate exposure latitude and depth of focus as determined by process window analysis. Inaccurate and/or imprecise measurement of the process window makes it difficult to achieve such improvements in throughput without jeopardizing device yield or device performance.

A further use of process window analysis is for process monitoring. Best focus, best exposure, exposure latitude and depth of focus can all be monitored over time using standard statistical techniques such as trend charting or statistical process control (SPC). Deviations between the current behavior of a charted variable and its historical behavior can be flagged for further investigation or other action. Inaccurate and/or imprecise measurement of the process window and its associated measurements reduces the effectiveness of trend charts or other forms of process control.

Additionally, the prior art geometric process window approach does not produce uncertainty estimates for various analysis outputs such as best focus, best exposure, depth of focus, and exposure latitude. Without uncertainty estimates, the usefulness of these outputs is reduced. For example, using the values of best dose and focus determined from process window analysis to control the scanner's dose and focus settings during manufacturing may result in changes in the uncertainty estimates for the process window best dose and/or focus. Anything larger should only result in changes to the scanner settings.

For these and other reasons, there is a need to discover a different method for process window measurement and analysis that solves one or more of the problems with prior art process window approaches.

15. Stochastic Process Window

A new method embodiment for process window measurement and analysis, and its use for process optimization and control will now be described. This new method embodiment, called PPW (Probabilistic Process Window), can solve many problems of the prior art methods (geometric process window, etc.) described above. This PPW approach can be used when two or more interacting process variables are varied to determine the impact on one or more output outcomes. In the main example used below, two process variables are discussed. Additionally, the main example below will use two process variables: exposure dose and focus during the process steps of lithographic patterning. However, the PPW approach is generally sufficient to handle more than two process variables handling lithography, etching, deposition, or other processing steps.

In some implementations, probabilistic process windows are used, among other things, to account for uncertainty in measurements to determine where the probabilities of representations of associations within a lithography process meet specification requirements. For example, instead of (or in addition to) ILM metrology tool 765, system 700 may include a probabilistic process window tool (not shown). The two-dimensional (2D) design of experiments (DoE) may include an sequentially increasing list of X values (possibly randomly spaced) representing a first DoE factor. This first DoE factor may be a process variable that has been intentionally changed to determine its effect on some output or outputs. The 2D DoE may also include a list of Y values (possibly randomly spaced) representing increasing second DoE factors. This second DoE factor may also be a process variable that has been intentionally changed to determine its effect on the output or outputs.

The 2D DoE may also include a set of specifications s (with minimum and/or maximum allowable values) that apply to some process response or output. For example, in a lithography process, one output may be the critical dimension (CD) of the printed feature. The specification for that output will describe the minimum allowable value for that output and/or the maximum allowable value for that output. As an example, the specification of the CD may be +/- 10% of the target (desired) value. For a target value of CD of 20 nm, the specifications (minimum and maximum, respectively) can be set to 18 nm and 22 nm.

some points For , 2D DoE also provides z-value (measurement result) and respective specifications may include the standard error (an expression of measurement uncertainty) corresponding to that z-value for . That is, measurement data can be represented by two three-dimensional arrays, one for z-values and one for standard errors. It is indexed with some points A process can be used to create a probabilistic process window to determine the probability that is within specifications. In some implementations, the process may include two main steps. In the first step, the probability that each given z-value is actually within specification s is determined using the standard error (or other measure of uncertainty) for that point, as explained further below. In the second stage, a bivariate probability distribution centered at some test points (x, y values) over the field generated in the first stage is x and y are applied to determine the probability that a test point is within specification, given some errors in y.

The experimental part of the PPW determination varies two or more process variables in the matrix. Two process variables (here referred to as x-input and y-input) are considered: exposure dose and focus. The exposure dose and focus are each varied over some range, for example, using a constant step size at the dose and a constant step size at the focus. Variable step sizes of dose and focus can also be used. A pattern is printed using this two-dimensional matrix of input values on one or more wafers. These wafers are then measured to determine one or more measurement outputs (herein referred to as z-outputs). For example, metrics of CD, unbiased roughness, defectivity, local CDU, pattern shape or fidelity, or other metrics of the output of the process may be measured.

In most cases, measurements of an output, such as CD, include an estimate of the measurement uncertainty for that output. For example, a single SEM image of a wafer may contain, for example, 20 lines and spaces. In such cases, it is common to produce as measured output the desired output the average CD, i.e. the average value of the CDs of each individual line on the image. The standard error of the mean can be used as an estimate of the measurement uncertainty of that mean CD. As is known in the field of statistics, other metrics of uncertainty of the mean may also be used. Sometimes multiples of the standard error (e.g., 2) are used as estimates of uncertainty. Other estimates of total measurement uncertainty may also be used.

Sometimes, for various reasons, one or more data points in the array of outputs as a function of input values may be missing. Therefore, before the first step of PPW determination, missing data points can be filled (their values are approximated), for example by interpolation. For example, Tables 5 and 6 below are measurements with missing data points. Other methods for handling missing data points may also be used.

Table 5

Mean line CD (Specification 1).

Table 6

Unbiased LER average (Specification 2).

A grid for each specification populated with as many values as can be determined, for example, by interpolation, to ensure that the output values in the first step of the process meet that specification (i.e., that the values are "within specifications"). “Awareness” or “in spec”) is determined. In the first part of the first step of the process, for each measured output (mean z-value) and the uncertainty for that output (e.g. σ _z , standard error of the mean z-value), and for each specification s For , the probability that the output is actually within specification is determined. For example, we can assume that the distribution of the output follows a normal (i.e. Gaussian) distribution. Other probability distributions are also possible. The calculation of the probability that the output is actually within specification will now be explained.

Given a mean output z, the standard error of that mean σ _z , and a specification s (minimum and maximum specification values, e.g. s _min and s _max ), we want to determine the probability that the measured mean value z is within the specification. For example, assuming a Gaussian distribution for the mean value z, the area under the normal curve is μ=z from s _min to s _max to determine the probability that the true value of z is [s _min , s _max ]. and σ=σ _z . An exemplary normal distribution cumulative distribution function (CDF) is given in Equation 11 below:

(11)

Here, erf is a well-known error function. The definition of the error function is given in Equation 12 below:

(12)

The probability that a z value is in spec can be determined using Equation 13 below:

(13)

If there is no maximum specification value, the first term is equal to 1. Additionally, if there is no minimum specification value, the second term is equal to 0. Figure 32 shows an example of interpreting the probability that a measurement value under the probability density curve is within specification as an appropriate area.

Without considering measurement uncertainty, in a traditional process window approach, the output value is either within specification or not. That is, a probability of 1 is validly attributed to an in-spec output, and a probability of 0 is validly attributed to an out-of-spec output.

In the second part of the first step, the probability that the output is within all specifications is determined. The output of the first stage may be called a “Probabilistic Process Window”. Given , for each point (pair of input values) (x,y), the probability that each point is within all specifications is determined. For example, based on the assumption that each P is independent (i.e., the measurement uncertainty for one output response is independent of the measurement uncertainty of the second output response), all probabilities for all specifications can be calculated using the equation below: are multiplied together as given in equation 14:

(14)

Next, within the specification |s| After starting with the probability grid, the final result of the first step is 1 grid of probabilities within specification. In some implementations, processor 755 of system 700 is configured to present a plurality of probabilities as one or more graphical elements of a graph displayed on a user interface (e.g., on output device 770). . For example, Figure 33 is an example of a heatmap showing the final result of the first step. The contour lines included in the heatmap illustrated in Figure 33 represent a traditional (geometric) process window and are shown for comparison purposes. An alternative view of the three-dimensional final result heat-map is illustrated in Figure 34.

In the second step of the process, using a grid of probabilities from the first step, a probability distribution using two variables of input values is gridded with some mean μ _x , μ _y and some error σ _x , σ _y Applies to. The average μ _x , μ _y represents the process set point, i.e. the nominal values of the input variables x and y used by the process. _{Errors σ} Often, two input variables x and y (such as focus and exposure dose) can be considered independent variables for purposes of generating a probability distribution using two variables, but this is not a requirement. . _A distribution with two variables, _with some _{mean μ}

In the first part of the second step, for each grid cell between the points, the volume under the center of the distribution using two variables at some mean μ _x , μ _y with some error σ _x , σ _y is determined. do. Next, the determined volume is weighted by the output of a function that is fit to the probabilities of the corner points of the current grid cell generated in the first step of the process. For example, using a Gaussian distribution, the volume can be determined using equation 14:

(14)

Next, all values in the grid are summed to determine the overall probability for this mean μ _x , μ _y with σ _x , σ _y . For example, all values in a grid can be summed using equation 15 below:

(15)

In the second part of the second step, the first part of the second step is repeated for different values of μ _x , μ _y to determine the in-spec probability with the mean centered on that point. . For example, using the original grid from the first step shown in Figures 33 and 34, each μ _x , μ _y value is determined, but μ _x , μ _y is set to the values of the original input data grid. Being is not a necessary requirement. Figure 35 is an |X| Each square illustrated in Figure 35 represents a bivariate Gaussian using two variables centered at that point. An alternative chart diagram to the chart shown in FIG. 35 is shown in FIG. 36.

In the third step of the process, a standard nonlinear optimization algorithm (e.g., Nelder-Mead or Gauss-Newton) determines μ with the highest probability for any error σ _x , σ _y . It is used to determine _x and μ _y . This represents the "best" value for the process settings for the input variables x and y. Here, “best” will mean the process setting _{(value μ x , μ y ) that produces} the highest fraction of functionality within specification given the errors σ For the FEM process window, this will be the best focus and best exposure. Setting the lithography tool to use this focus and exposure dose can produce the best printing results.

In the fourth step of the process, a σ _y versus σ _x curve is generated. A step size of σ _x is chosen. _{As σ _} indicates a low point (e.g. 99.73%). The increment of σ _y is stopped if the cutoff criterion is not met. The determined points are plotted to produce a σ _y versus σ _x curve. In some implementations, σ _x is depth of focus and σ _y is exposure latitude. Figure 37 is an example of a σ _y versus σ _x curve. Traditional process window analysis can be based on finding the largest rectangle or ellipse that geometrically fits inside the process window. Similar but more stringent curves can be generated using a Probabilistic Process Window using the steps described above.

Once a stochastic process window is determined for a particular dataset with certain specifications, the stochastic process window can be combined with any number of other stochastic process windows, as long as two or more input process variables are identical and have similar bounds. It can be. Different stochastic process windows may have different default output values, feature types, and specifications and still be combined, due to the unit-less nature of probabilities. To combine the stochastic process windows, each test point can be evaluated and the resulting probabilities are multiplied together to determine the combined probability across the stochastic process windows.

Referring now to FIG. 38, an example method 3800 is provided for generating a probabilistic process window that accounts for measurement uncertainty. The method 3800 begins (block 3802) and selects (3804) a first variable displayed on the first axis of the graph. Next, the method 3800 proceeds to select a second variable displayed on the second axis of the graph (block 3806). In some implementations, the first variable and the second variable are associated with a stochastic process window. In some implementations, the first variable includes the exposure dose and the second variable includes the focus of the scanning lithography process. In alternative implementations, the first variable includes etch time and the second variable includes wafer temperature. Many different combinations of process variables are possible.

Next, the method 3800 proceeds to select at least one response variable (or output response variable) that is a function of the first variable and the second variable (block 3808). These output response variables are measured as a function of first and second variables. Each output response variable can be graphed as a function of first and second variables. For each output response variable, specifications are set for values of the output response that are considered acceptable.

Next, the method 3800 proceeds to determine the measurement uncertainty for each of the output response variables (block 3810). Next, method 3800 determines, based on the measurements of one or more output response variables and the measurement uncertainty for at least one output response variable, that a plurality of points associated with the lithography process meet the specification requirements for each output response variable. Proceed to determine a plurality of probabilities representing the plurality of indications (block 3812). A plurality of probabilities represent a process window. Next, method 3800 proceeds to configure a lithography tool for manufacturing a semiconductor device, based on the process window.

In some implementations, a third input variable is selected. In some implementations, the third variable is temperature. Additionally, a second plurality of points associated with the lithographic process meet specification requirements based on a measurement of the output response as a function of a third variable combined with variations of the first and second variables. A second plurality of probabilities for expressing the indications are determined. The second plurality of probabilities represent a second process window.

The plurality of probabilities can be modified to achieve the desired cost optimization of manufacturing a semiconductor device. For example, in some implementations, one or more graphical elements are presented configured to allow a user to change a first and/or second variable to modify a plurality of probabilities in real time or near real time. Real-time can mean less than 2 seconds. Near real-time may refer to any interaction of sufficiently brief duration to allow two individuals to engage in conversation through such user interface, typically less than 10 seconds (or any suitable proximity difference between two different times). ), but it will be over 2 seconds.

In some embodiments, process range variations are selected for the first and second variables. Given the settings and process range for each of the first and second variables, the fraction of features that meet the specification requirements is determined. Based on the fraction of features that meet the specification requirements, a determination may be made regarding settings of the first and second variables that produce the maximum fraction of features that meet the specification.

16. Process control using stochastic process windows

Controlling focus and dose is an important part of keeping the critical dimensions of the printed pattern under control during semiconductor manufacturing. The first step in understanding how to control focus and dose is to characterize the response of critical features to variations using the focus-exposure process window, as discussed above. Proper analysis of CD (and other output responses) versus focus and dose data allows calculation of the process window, measurement of the process window size to generate exposure latitude-DOF plots, and determination of a single value for depth of focus. . As discussed above, the best analysis methods use stochastic process windows.

The best focus and dose are also determined using this analysis as process settings that maximize tolerance for focus and dose errors based on PPW. Once the best focus and dose are determined, the next goal is to ensure that the process is centered on these best focus and dose conditions as the production wafers pass through the lithography cell. Since almost all errors that occur in lithography act as effective dose error or effective focus error, adjusting the dose and focus appropriately after an appropriate interval will provide much tighter patterning control. You can. Ultimately, improved patterning control can lead to higher manufacturing yields and better performance of the devices being manufactured.

Often, the variables dose and focus, as well as other process variables that can be used in stochastic process window analysis, are monitored and controlled using the well-known Advanced Process Control (APC) methodology. These APC methods include feed-forward and feedback loops to control process tools. This process control method will be enhanced by replacing the traditional process window analysis with the more rigorous and accurate PPW analysis described above.

Process window analysis can also be used for wafer placement purposes. Wafer Placement takes the output of the process window analysis and determines whether the wafers represented by the process window should proceed to subsequent processes or be rejected due to expected poor printed patterns. If rejected, these wafers can be reprocessed or discarded. Inaccurate decisions during wafer placement can be very costly, delivering bad wafers for subsequent processing, or rejecting good wafers. For example, improved accuracy in process window analysis using the PPW process described above can improve wafer placement accuracy.

One approach to process control uses machine learning. A machine learning algorithm will attempt to make predictions related to, for example, yield or device performance based on a number of process variables used in the manufacture of devices. A predictive machine learning model is first trained using measured outputs (things to be predicted) as a function of measured inputs (process variables and intermediate results, such as the CD of a particular feature at a particular stage of the manufacturing process). Machine learning algorithms can use process window information for both training and prediction. As such, the improved accuracy and precision of the PPW approach can result in improved machine learning predictions.

Although the above-described implementations reference top-down images of nominally planar pattern structures to measure edge roughness, the present disclosure is not limited to such pattern structure geometries. The present invention can be used to measure three-dimensional, non-planar, curved, or tilted structures. In addition to edge roughness, surface roughness can be measured and analyzed using techniques similar to those described in this disclosure.

Although the above-described implementations refer to measurements of roughness, the present disclosure can be used to make other measurements as well. For example, highly accurate determination of pattern structure edges can be used for measurement of feature width, feature placement, edge placement, and other similar measurements. Contours of measured features can be used for many purposes, such as modeling or controlling the performance of the measured device. By collecting measurements from many samples and averaging them statistically, much greater accuracy (lower uncertainty) can be obtained.

In accordance with the above disclosure, the examples of systems and methods listed in the following items are considered specifically and are intended as a non-limiting set of examples.

Item 1. A computer implemented method comprising:

selecting a first variable;

selecting a second variable;

selecting at least one response variable that is a function of the first variable and the second variable;

determining measurement uncertainty for each response variable;

A plurality of probabilities representing a plurality of indications of whether a plurality of points associated with a lithographic process meet specification requirements for each response variable, based on a measurement of the response variable and a measurement uncertainty for the response variable. determining the probabilities, wherein the plurality of probabilities represent a process window; and

Based on the process window, a computer-implemented method comprising configuring a lithography tool for manufacturing a semiconductor device.

Item 2. The method of clause 1, wherein the configuring step sets a first operating parameter based on the first variable and sets a second operating parameter based on the second variable. A computer implemented method comprising transmitting.

Item 3. The computer-implemented method of any item herein, further comprising presenting the plurality of probabilities as graphical elements of a graph on a user interface of the computing device.

Item 4. The computer-implemented method of any clause herein, wherein the presentation of the plurality of probabilities is a heatmap or a 3D plot or a contour plot on a user interface.

Item 5. The computer-implemented method of any clause herein, wherein the measurement uncertainty of one or more response variables is expressed by a Gaussian normal probability distribution.

Clause 6. The computer-implemented method of any clause herein, wherein the first and second variables are associated with a stochastic process window.

Clause 7. The computer-implemented method of any clause herein, wherein the first variable comprises an exposure dose and the second variable comprises a focus of a scanning lithography process.

Item 8. The clause of any item herein, wherein one is configured to allow a user to change properties of the first variable, the second variable, or both to modify the plurality of probabilities in real time or near real time. A computer-implemented method, further comprising presenting the above graphical elements.

Item 9. The method of any item herein, configured to allow a user to change characteristics of the first variable, the second variable, or both to modify the plurality of probabilities in real time or near real time. A computer-implemented method further comprising presenting one or more graphical elements, wherein modifying the plurality of probabilities is associated with a desired cost optimization of manufacturing the semiconductor device.

Item 10. In the system,

lithography tools;

a memory device that stores instructions; and

comprising a processing device coupled to the memory device and the lithography tool, wherein the processing device executes instructions,

select a first variable;

select a second variable;

select a response variable dependent on the first variable and the second variable;

*Determine measurement uncertainty for the response variable;

Based on a measurement of the response variable and a measurement uncertainty for the response variable, determine a plurality of probabilities indicating a plurality of indications of whether a plurality of points associated with a lithographic process meet specification requirements for each response variable, and - The plurality of probabilities represent a process window; and

Based on the process window, the system configures the lithography tool to fabricate a semiconductor device.

Item 11. The method of any item herein, wherein the configuring controls the lithography tool to set a first operating parameter based on the first variable and set a second operating parameter based on the second variable. A system comprising transmitting signals.

Item 12. The system of any item herein, wherein the processing device is further configured to present the plurality of probabilities as a graphical element of the graph on a user interface of the computing device.

Item 13. The system of any item herein, wherein the presentation of the plurality of probabilities is a heat-map on the user interface.

Item 14. The system of any item herein, wherein the measurement uncertainty of the response variable is a Gaussian normal probability distribution.

Item 15. The system of any item herein, wherein the first and second variables are associated with a stochastic process window.

Item 16. The system of any item herein, wherein the first variable comprises an exposure dose and the second variable comprises a focus of a scanning lithography process.

Item 17. The method of any item herein, wherein the processing device:

select process range variations for the first and second variables;

Given settings and process ranges for each of the first and second variables, determine the fraction of features that meet the specification requirements;

The system further configured to determine settings of the first and second variables that produce a maximum fraction of features that meet a specification based on the fraction of features that meet a specification requirement.

Item 18. A tangible, non-transitory computer-readable medium storing instructions, wherein when the instructions are executed, a processing device

select a first variable;

select a second variable;

select at least one response variable that is a function of the first variable and the second variable;

determine measurement uncertainty for each of the response variables;

Based on a measurement of the response variable and a measurement uncertainty for the response variable, determine a plurality of probabilities indicating a plurality of indications of whether a plurality of points associated with a lithographic process meet specification requirements for each response variable, and - A plurality of probabilities represent a process window -; and

A computer-readable medium that allows configuring a lithography tool to fabricate a semiconductor device based on the process window.

Item 19. The method of any item herein, wherein the configuring comprises: setting a first operating parameter based on the first variable, and setting a second operating parameter based on the second variable, to the lithography tool. A computer-readable medium comprising transmitting control signals.

Item 20. The computer-readable medium of any clause herein, wherein the processing device is further configured to present the plurality of probabilities as graphical elements of the graph on a user interface of the computing device.

The terminology used herein is only used to describe specific implementations and is not intended to limit the disclosure. As used herein, the singular forms “a”, “an” and “the” are intended to include plural forms as well, unless the context clearly dictates otherwise. As used herein, the terms “comprises” and/or “comprising” specify the presence of a referenced feature, integer, step, operation, element and/or component, but may also include one or more other features. , it will be further understood that it does not exclude the presence or addition of integers, steps, operations, elements, components and/or groups thereof.

Note that not all of the activities described above may be required in the overall description or examples, some of the specific activities may not be required, and one or more additional activities may be performed in addition to those described. Additionally, the order in which activities are listed is not necessarily the order in which they will be performed.

It may be advantageous to provide definitions of certain words and phrases used throughout this patent document. The term “communication” as well as its derivatives include both direct and indirect communication. The terms “include” and “comprise” as well as their derivatives mean inclusion without limitation. The term “or” refers generically to and/or. The phrase “associate” and its derivatives include, included within, interconnect, contain, be contained within, connect, combine, communicable, cooperate, intervene, juxtapose, proximate, bind. It means to become, to possess, to have attributes, to form relationships, etc. The phrase “at least one”, when used with a list of items, means that different combinations of one or more of the listed items may be used and that only one item in the list may be needed. For example, “at least one of A, B, and C” includes any of the following combinations: A, B, C, A and B, A and C, B and C, and A and B and C.

The description herein should not be read to imply that any particular element, step or function is essential or critical to be included in the scope of the claims. The scope of patented subject matter is defined solely by the permitted claims. Moreover, no claim shall be permitted in any of the appended claims or claim elements unless the precise words "means" or "step" are specifically used in a subsequent particular claim to identify a function. 35 U.S.C. Do not invoke § 112(f). “mechanism,” “module,” “device,” “unit,” “component,” “element,” “member,” “device,” “machine,” “system,” “processor,” or “controller” in the claims; Use of the same (but not limited) terms is understood and intended to refer to structures known to those skilled in the art, as further modified or improved by the features of the claims themselves, and as defined in 35 U.S.C. § 112(f) is not intended to be invoked.

Advantages, other advantages, and solutions to problems are described above with respect to specific implementations. However, the advantages, advantages, solutions to problems, and any feature(s) that would cause any advantage, advantage, or solution to arise or become more pronounced are not covered by any or all of the claims. It should not be construed as a critical, required, or essential feature.

After reading this specification, one skilled in the art will recognize that, for clarity, certain features described herein in the context of separate implementations may also be provided in combination in a single embodiment. Conversely, for the sake of brevity, various features described in the context of a single embodiment could also be provided individually or in any sub-combination. Additionally, references to values stated in ranges include each and every value within that range.

Corresponding structures, materials, operations, and equivalents of all means or steps plus functional elements in the claims below include any structure for performing the function in combination with other specifically claimed elements; It is intended to include materials or actions. The description of the invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the disclosure. The implementations have been selected and described to best explain the principles and practical application of the disclosure and to enable those skilled in the art to understand the disclosure in various implementations with various modifications suitable for the particular use contemplated.

Claims (20)

1. A computer implemented method, comprising:
selecting a first variable;
selecting a second variable;
selecting at least one response variable that is a function of the first variable and the second variable;
determining measurement uncertainty for each response variable;
a plurality of indications of whether a plurality of points associated with a lithographic process meet specification requirements for each response variable, based on the measurement of the response variable and the measurement uncertainty for the response variable. determining probabilities, the plurality of probabilities representing a process window; and
Based on the process window, configuring a lithography tool for manufacturing a semiconductor device.
Including a computer implemented method. According to paragraph 1,
The configuring step includes sending control signals to the lithography tool to set a first operating parameter based on the first variable and set a second operating parameter based on the second variable. How to implement it. According to paragraph 1,
The computer-implemented method further comprising presenting the plurality of probabilities as graphical elements of a graph on a user interface of a computing device. According to paragraph 3,
The computer-implemented method of claim 1, wherein the presentation of the plurality of probabilities is a heat-map, three-dimensional plot, or contour plot on the user interface. According to paragraph 1,
A computer-implemented method, wherein the measurement uncertainty of the response variable is expressed as a Gaussian normal probability distribution. According to paragraph 1,
wherein the first variable is represented on a first axis of the graph and the second variable is represented on a second axis of the graph. The computer-implemented method of claim 1, wherein the first variable includes an exposure dose and the second variable includes a focus of a scanning lithography process. According to paragraph 1,
further comprising presenting one or more graphical elements configured to enable a user to change properties of the first variable, the second variable, or both to modify the plurality of probabilities in real time or near real time. Computer implementation method. According to paragraph 1,
further comprising presenting one or more graphical elements configured to enable a user to change properties of the first variable, the second variable, or both to modify the plurality of probabilities in real time or near real time; ,
Wherein modifying the plurality of probabilities is associated with a desired cost optimization of manufacturing the semiconductor device. As a system,
lithography tools;
a memory device that stores instructions; and
A processing device coupled to the memory device and the lithography tool.
Including,
The processing device is,
select a first variable;
select a second variable;
select response variables dependent on the first variable and the second variable;
determine measurement uncertainty for the response variable;
Based on a measurement of the response variable and a measurement uncertainty for the response variable, determine a plurality of probabilities indicating a plurality of indications of whether a plurality of points associated with a lithographic process meet specification requirements for each response variable, and - The plurality of probabilities represent a process window; and
Based on the process window, configuring the lithography tool for manufacturing a semiconductor device.
A system that executes commands. According to clause 10,
The configuring includes setting a first operating parameter based on the first variable and sending control signals to the lithography tool to set a second operating parameter based on the second variable. According to clause 10,
wherein the processing device is further configured to present the plurality of probabilities as graphical elements of a graph on a user interface of the computing device. According to clause 12,
The system of claim 1, wherein the presentation of the plurality of probabilities is a heat-map on the user interface. 11. The system of claim 10, wherein the measurement uncertainty of the response variable is a Gaussian normal probability distribution. According to clause 10,
The system of claim 1, wherein the first variable is represented on a first axis of the graph and the second variable is represented on a second axis of the graph. According to clause 10,
The system of claim 1, wherein the first variable includes an exposure dose and the second variable includes a focus of the scanning lithography process. According to clause 10,
The processing device is,
select process range variations for the first variable and the second variable;
Determine the fraction of features for which settings and process ranges for each of the first and second variables meet given specification requirements,
The system is further configured to determine, based on the fraction of features that meet specification requirements, settings of the first variable and the second variable that produce a maximum fraction of features that meet a specification. A tangible, non-transitory computer-readable medium storing instructions, comprising:
The above instructions, when executed, cause the processing device to:
select a first variable;
select a second variable;
select at least one response variable that is a function of the first variable and the second variable;
Determine the measurement uncertainty for each response variable;
Based on a measurement of the response variable and a measurement uncertainty for the response variable, determine a plurality of probabilities indicating a plurality of indications of whether a plurality of points associated with a lithographic process meet specification requirements for each response variable, and - The plurality of probabilities represent a process window; and
A computer-readable medium that allows configuring a lithography tool for manufacturing a semiconductor device based on the process window. According to clause 18,
wherein configuring includes sending control signals to the lithography tool to set a first operating parameter based on the first variable and set a second operating parameter based on the second variable. media. According to clause 18,
wherein the processing device is further configured to present the plurality of probabilities as graphical elements of a graph on a user interface of a computing device.